DP Mathematics SL Questionbank

• Syllabus

Topic 6 - Calculus

Sub sections and their related questions

6.1

• 12N.1.sl.TZ0.4a: Find \(f'(x)\) .
• 12N.1.sl.TZ0.4b: The graph of \(f\) has a gradient of \(3\) at the point P. Find the value of \(a\) .
• 08N.2.sl.TZ0.9b: Let the line L be the normal to the curve of f at \(x = 0\) . Find the equation of L .
• 08M.1.sl.TZ1.8d: The line L is the tangent to the curve of f at \((3{\text{, }}12)\). Find the equation of L in...
• 08M.2.sl.TZ2.9d: Let L be the normal to the curve of f at \({\text{P}}(0{\text{, }}1)\) . Show that L has equation...
• 12M.1.sl.TZ1.3a: Write down \(f'(x)\) .
• 12M.1.sl.TZ1.3b(i) and (ii): The tangent to the graph of f at the point \({\text{P}}(0{\text{, }}b)\) has gradient m...
• 12M.1.sl.TZ1.3c: Hence, write down the equation of this tangent.
• 10N.1.sl.TZ0.2a: Find \(g'(x)\) .
• 10N.1.sl.TZ0.2b: Find the gradient of the graph of g at \(x = \pi \) .
• 10N.1.sl.TZ0.10a(i), (ii) and (iii): (i)     Show that the gradient of [PQ] is \(\frac{{{a^3}}}{{a - \frac{2}{3}}}\) . (ii)    Find...
• 10N.1.sl.TZ0.10b: Given that the area of T is \(2k + 4\) , show that k satisfies the equation...
• 10M.1.sl.TZ2.5: Let \(f(x) = k{x^4}\) . The point \({\text{P}}(1{\text{, }}k)\) lies on the curve of f . At P,...
• 09N.1.sl.TZ0.10a: Show that the equation of L is \(y = - 4x + 18\) .
• 09N.2.sl.TZ0.5: Consider the curve with equation \(f(x) = p{x^2} + qx\) , where p and q are constants. The point...
• 09M.1.sl.TZ1.3: Let \(f(x) = {{\rm{e}}^x}\cos x\) . Find the gradient of the normal to the curve of f at...
• 09M.2.sl.TZ1.10b: Use the formula \(f'(x) = \mathop {\lim }\limits_{h \to 0} \frac{{f(x + h) - f(x)}}{h}\) to show...
• 09M.2.sl.TZ1.10c: The tangent to the curve of f at the point \({\text{P}}(1{\text{, }} - 2)\) is parallel to the...
• 09M.2.sl.TZ1.10d: The graph of f is decreasing for \(p < x < q\) . Find the value of p and of q.
• 09M.2.sl.TZ2.6a: Write down the gradient of the curve at P.
• 09M.2.sl.TZ2.6b: The normal to the curve at P cuts the x-axis at R. Find the coordinates of R.
• 10M.2.sl.TZ1.9a: Show that \(A = 10\) .
• 10M.2.sl.TZ1.9b: Given that \(f(15) = 3.49\) (correct to 3 significant figures), find the value of k.
• 10M.2.sl.TZ1.9c(i), (ii) and (iii): (i)     Using your value of k , find \(f'(x)\) . (ii)    Hence, explain why f is a decreasing...
• 10M.2.sl.TZ1.9d: Let \(g(x) = - {x^2} + 12x - 24\) . Find the area enclosed by the graphs of f and g .
• SPNone.2.sl.TZ0.2c: Write down the gradient of the graph of f at \(x = 3\) .
• 11N.1.sl.TZ0.9a(i), (ii) and (iii): Use the graph to write down the value of (i)     a ; (ii)    c ; (iii)   d .
• 11N.1.sl.TZ0.9b: Show that \(b = \frac{\pi }{4}\) .
• 11N.1.sl.TZ0.9c: Find \(f'(x)\) .
• 11N.1.sl.TZ0.9d: At a point R, the gradient is \( - 2\pi \) . Find the x-coordinate of R.
• 11N.1.sl.TZ0.10a: Find the equation of L .
• 11N.1.sl.TZ0.10b: Find the area of the region enclosed by the curve of g , the x-axis, and the lines \(x = 2\) and...
• 11N.1.sl.TZ0.10c: The graph of g is reflected in the x-axis to give the graph of h . The area of the region...
• 11N.2.sl.TZ0.10a: Sketch the graph of f .
• 11N.2.sl.TZ0.10b(i) and (ii): (i)     Write down the x-coordinate of the maximum point on the graph of f . (ii)    Write down...
• 11N.2.sl.TZ0.10c: Show that \(f'(x) = \frac{{20 - 6x}}{{{{\rm{e}}^{0.3x}}}}\) .
• 11N.2.sl.TZ0.10d: Find the interval where the rate of change of f is increasing.
• 11M.2.sl.TZ1.8a: Show that \(a = 4\) .
• 11M.2.sl.TZ1.8b: The wheel turns at a rate of one rotation every 30 seconds. Show that \(b = \frac{\pi }{{15}}\) .
• 11M.2.sl.TZ1.8c: In the first rotation, there are two values of t when the bucket is descending at a rate of...
• 11M.2.sl.TZ1.8d: In the first rotation, there are two values of t when the bucket is descending at a rate of...
• 11M.1.sl.TZ2.8a: Show that the equation of T is \(y = 4x - 2\) .
• 11M.1.sl.TZ2.8b: Find the x-intercept of T .
• 11M.1.sl.TZ2.8c(i) and (ii): The shaded region R is enclosed by the graph of f , the line T , and the x-axis. (i)     Write...
• 13M.1.sl.TZ1.3b: Find the gradient of the curve of \(f\) at \(x = \frac{\pi }{2}\) .
• 13M.1.sl.TZ1.10b: Find the set of values of \(x\) for which \(f\) is increasing.
• 13M.2.sl.TZ1.9e: Find the maximum rate of change of \(f\) .
• 13M.1.sl.TZ2.9d: Find the value of \(x\) for which the tangent to the graph of \(f\) is parallel to the tangent to...
• 13M.1.sl.TZ2.10d: find the equation of the normal to the graph of \(h\) at P.
• 13M.2.sl.TZ2.10b: Consider all values of \(m\) such that the graphs of \(f\) and \(g\) intersect. Find the value of...
• 14M.2.sl.TZ1.5b(i): Find the rate of change of the deer population on 1 May 2014.
• 14M.2.sl.TZ1.5b(ii): Interpret the answer to part (i) with reference to the deer population size on 1 May 2014.
• 14M.2.sl.TZ1.7: Let \(f(x) = \frac{{g(x)}}{{h(x)}}\), where \(g(2) = 18,{\text{ }}h(2) = 6,{\text{ }}g'(2) = 5\),...
• 13N.1.sl.TZ0.6: Let \(f(x) = {{\text{e}}^{2x}}\). The line \(L\) is the tangent to the curve of \(f\) at...
• 13N.2.sl.TZ0.7b(i): Find the rate of change of area when \(x = 2\).
• 13N.2.sl.TZ0.7b(ii): The area is decreasing for \(a < x < b\). Find the value of \(a\) and of \(b\).
• 15M.1.sl.TZ1.9d: Find the equation of the tangent to the curve of \(f\) at \(( - 2,{\text{ }}1)\), giving your...
• 15M.2.sl.TZ1.6: Let \(f(x) = \frac{{\ln (4x)}}{x}\) for \(0 < x \le 5\). Points...
• 15N.1.sl.TZ0.10b: Find the set of values of \(x\) for which the graph of \(f\) is concave down.
• 15N.1.sl.TZ0.10d: The following diagram shows the shaded regions \(A\), \(B\) and \(C\). The regions are...
• 16M.1.sl.TZ1.10a: Find \(f'(1)\).
• 16M.1.sl.TZ1.10b: Write down \(g'(1)\).
• 16M.1.sl.TZ1.10c: Find \(g(1)\).
• 16M.1.sl.TZ1.10d: Let \(h(x) = f(x) \times g(x)\). Find the equation of the tangent to the graph of \(h\) at the...
• 16M.1.sl.TZ2.10a: (i)     Given that \(f'(x) = \frac{{2{a^2} - 4{x^2}}}{{\sqrt {{a^2} - {x^2}} }}\), for...
• 16M.1.sl.TZ2.10b: Show that \({A_R} = \frac{2}{3}{a^3}\).
• 16M.1.sl.TZ2.10c: Let \({A_T}\) be the area of the triangle OPQ. Given that \({A_T} = k{A_R}\), find the value of...
• 16M.2.sl.TZ2.9a: Write down the equation of the horizontal asymptote of the graph of \(f\).
• 16M.2.sl.TZ2.9b: Find \(f'(x)\).
• 16M.2.sl.TZ2.9c: Write down the value of \(b\).
• 16M.2.sl.TZ2.9d: Given that \(g'(1) =  - e\), find the value of \(a\).
• 16M.2.sl.TZ2.9e: There is a value of \(x\), for \(1 < x < 4\), for which the graphs of \(f\) and \(g\) have...
• 16N.2.sl.TZ0.2b: (i)     sketch the graph of \(f\), clearly indicating the point A; (ii)    sketch the tangent to...
• 16N.2.sl.TZ0.10a: (i)     Find the value of \(c\). (ii)     Show that \(b = \frac{\pi }{6}\). (iii)     Find the...
• 16N.2.sl.TZ0.10b: (i)     Write down the value of \(k\). (ii)     Find \(g(x)\).
• 16N.2.sl.TZ0.10c: (i)     Find \(w\). (ii)     Hence or otherwise, find the maximum positive rate of change of \(g\).
• 17M.1.sl.TZ1.6a.i: Write down the gradient of the curve of \(f\) at P.
• 17M.1.sl.TZ1.6a.ii: Find the equation of the normal to the curve of \(f\) at P.
• 17M.1.sl.TZ1.9c: The line \(y = kx - 5\) is a tangent to the curve of \(f\). Find the values of \(k\).
• 17M.1.sl.TZ2.10a.i: Write down \(f'(x)\).
• 17M.1.sl.TZ2.10a.ii: Find the gradient of \(L\).
• 17M.1.sl.TZ2.10b: Show that the \(x\)-coordinate of B is \( - \frac{k}{2}\).
• 17M.1.sl.TZ2.10c: Find the area of triangle ABC, giving your answer in terms of \(k\).
• 17M.1.sl.TZ2.10d: Given that the area of triangle ABC is \(p\) times the area of \(R\), find the value of \(p\).
• 17M.2.sl.TZ2.8a: Find the value of \(p\).
• 17M.2.sl.TZ2.8b.i: Write down the coordinates of A.
• 17M.2.sl.TZ2.8b.ii: Write down the rate of change of \(f\) at A.
• 17M.2.sl.TZ2.8c.i: Find the coordinates of B.
• 17M.2.sl.TZ2.8c.ii: Find the the rate of change of \(f\) at B.
• 17M.2.sl.TZ2.8d: Let \(R\) be the region enclosed by the graph of \(f\) , the \(x\)-axis, the line \(x = b\) and...
• 17N.1.sl.TZ0.5a: Find \((g \circ f)(x)\).
• 17N.1.sl.TZ0.5b: Given that \(\mathop {\lim }\limits_{x \to  + \infty } (g \circ f)(x) =  - 3\), find the value of...
• 17N.1.sl.TZ0.8a: Show that \(f’(1) = 1\).
• 17N.1.sl.TZ0.8b: Find the equation of \(L\) in the form \(y = ax + b\).
• 17N.1.sl.TZ0.8d: Find the area of the region enclosed by the graph of \(f\) and the line \(L\).
• 17N.1.sl.TZ0.8c: Find the \(x\)-coordinate of Q.
• 18M.1.sl.TZ1.7: Consider f(x), g(x) and h(x), for x∈\(\mathbb{R}\) where h(x) = \(\left( {f \circ g}...
• 18M.1.sl.TZ2.10a.i: Write down \(f'\left( 2 \right)\).
• 18M.1.sl.TZ2.10a.ii: Find \(f\left( 2 \right)\).
• 18M.1.sl.TZ2.10b: Show that the graph of g has a gradient of 6 at P.
• 18M.1.sl.TZ2.10c: Let L2 be the tangent to the graph of g at P. L1 intersects L2 at the point Q. Find the...

6.2

• 12N.1.sl.TZ0.4a: Find \(f'(x)\) .
• 12N.1.sl.TZ0.4b: The graph of \(f\) has a gradient of \(3\) at the point P. Find the value of \(a\) .
• 12N.1.sl.TZ0.10a: Find \(f'(x)\) .
• 12N.1.sl.TZ0.10b: Let \(g(x) = \ln \left( {\frac{{6x}}{{x + 1}}} \right)\) , for \(x > 0\) . Show that...
• 12M.1.sl.TZ2.10a: Use the quotient rule to show that \(f'(x) = \frac{{2{x^2} - 2}}{{{{( - 2{x^2} + 5x - 2)}^2}}}\) .
• 12M.1.sl.TZ2.10b: Hence find the coordinates of B.
• 12M.1.sl.TZ2.10c: Given that the line \(y = k\) does not meet the graph of f , find the possible values of k .
• 12M.2.sl.TZ2.2a: Find \(f'(x)\) .
• 12M.2.sl.TZ2.2b: On the grid below, sketch the graph of \(f'(x)\) .
• 12M.2.sl.TZ2.9a: Show that \(8a + 4b + c = 9\) .
• 12M.2.sl.TZ2.9b: The graph of f has a local minimum at \((1{\text{, }}4)\) . Find two other equations in a , b...
• 12M.2.sl.TZ2.9c: Find the value of a , of b and of c .
• 12N.1.sl.TZ0.10c: Let \(h(x) = \frac{1}{{x(x + 1)}}\) . The area enclosed by the graph of h , the x-axis and the...
• 08N.2.sl.TZ0.9a: Show that \(f'(x) = {{\rm{e}}^{2x}}(2\cos x - \sin x)\) .
• 08M.1.sl.TZ1.8a: Find \(f'(x)\) .
• 08M.2.sl.TZ1.10b(i) and (ii): (i)     Find \(f'(x)\) . (ii)    Find \(g'(x)\) .
• 08M.1.sl.TZ2.9b: Find \(f'(x)\) , giving your answer in the form \(a{\sin ^p}x{\cos ^q}x\) where...
• 08M.2.sl.TZ2.9a: Show that \(f'(x) = {{\rm{e}}^x}(1 - 2x - {x^2})\) . 
• 12M.1.sl.TZ1.3a: Write down \(f'(x)\) .
• 12M.1.sl.TZ1.3b(i) and (ii): The tangent to the graph of f at the point \({\text{P}}(0{\text{, }}b)\) has gradient m...
• 12M.1.sl.TZ1.3c: Hence, write down the equation of this tangent.
• 12M.1.sl.TZ1.10a: Find \(s'(t)\) .
• 12M.1.sl.TZ1.10b: In this interval, there are only two values of t for which the object is not moving. One value is...
• 12M.1.sl.TZ1.10c: Show that \(s'(t) > 0\) between these two values of t .
• 12M.1.sl.TZ1.10d: Find the distance travelled between these two values of t .
• 10N.1.sl.TZ0.2a: Find \(g'(x)\) .
• 10N.1.sl.TZ0.2b: Find the gradient of the graph of g at \(x = \pi \) .
• 10N.1.sl.TZ0.10a(i), (ii) and (iii): (i)     Show that the gradient of [PQ] is \(\frac{{{a^3}}}{{a - \frac{2}{3}}}\) . (ii)    Find...
• 10N.1.sl.TZ0.10b: Given that the area of T is \(2k + 4\) , show that k satisfies the equation...
• 10M.1.sl.TZ1.8a: Find the coordinates of A.
• 10M.1.sl.TZ1.8b(i), (ii) and (iii): Write down the coordinates of (i)     the image of B after reflection in the y-axis; (ii)   ...
• 10M.1.sl.TZ1.9a: Use the quotient rule to show that \(f'(x) = \frac{{ - 1}}{{{{\sin }^2}x}}\) .
• 10M.1.sl.TZ1.9b: Find \(f''(x)\) .
• 10M.1.sl.TZ1.9c: Find the value of p and of q.
• 10M.1.sl.TZ1.9d: Use information from the table to explain why there is a point of inflexion on the graph of f...
• 10M.1.sl.TZ2.5: Let \(f(x) = k{x^4}\) . The point \({\text{P}}(1{\text{, }}k)\) lies on the curve of f . At P,...
• 09N.1.sl.TZ0.5a: Find \(f'(x)\) .
• 09N.1.sl.TZ0.9a: (i)     Find the coordinates of A. (ii)    Show that \(f'(x) = 0\) at A.
• 09N.2.sl.TZ0.2a: Find \(f'(x)\) .
• 09N.2.sl.TZ0.2b: Find \(g'(x)\) .
• 09N.2.sl.TZ0.2c: Let \(h(x) = f(x) \times g(x)\) . Find \(h'(x)\) .
• 09M.1.sl.TZ1.3: Let \(f(x) = {{\rm{e}}^x}\cos x\) . Find the gradient of the normal to the curve of f at...
• 09M.1.sl.TZ1.8c: (i)     Find \(\frac{{{\rm{d}}A}}{{{\rm{d}}\theta }}\) . (ii)    Hence, find the exact value of...
• 09M.1.sl.TZ2.6a: Find the second derivative.
• 09M.1.sl.TZ2.6b: Find \(f'(3)\) and \(f''(3)\) .
• 09M.1.sl.TZ2.8a: Write down (i)     \(f'(x)\) ; (ii)    \(g'(x)\) .
• 09M.1.sl.TZ2.8b: Let \(h(x) = {{\rm{e}}^{ - 3x}}\sin \left( {x - \frac{\pi }{3}} \right)\) . Find the exact value...
• 09M.2.sl.TZ2.6a: Write down the gradient of the curve at P.
• 09M.2.sl.TZ2.10d: Write down one value of x such that \(f'(x) = 0\) .
• 10M.2.sl.TZ1.3a: Find \(f'(x)\) .
• 10M.2.sl.TZ1.3b: On the grid below, sketch the graph of \(y = f'(x)\) .
• 10M.2.sl.TZ1.9a: Show that \(A = 10\) .
• 10M.2.sl.TZ1.9b: Given that \(f(15) = 3.49\) (correct to 3 significant figures), find the value of k.
• 10M.2.sl.TZ1.9c(i), (ii) and (iii): (i)     Using your value of k , find \(f'(x)\) . (ii)    Hence, explain why f is a decreasing...
• 10M.2.sl.TZ1.9d: Let \(g(x) = - {x^2} + 12x - 24\) . Find the area enclosed by the graphs of f and g .
• 10M.2.sl.TZ2.10a(i) and (ii): Let P and Q be points on the curve of f where the tangent to the graph of f is parallel to the...
• 10M.2.sl.TZ2.10b: Let \(g(x) = {x^3}\ln (4 - {x^2})\) , for \( - 2 < x < 2\) . Show that...
• 10M.2.sl.TZ2.10c: Let \(g(x) = {x^3}\ln (4 - {x^2})\) , for \( - 2 < x < 2\) . Sketch the graph of \(g'\) .
• 10M.2.sl.TZ2.10d: Let \(g(x) = {x^3}\ln (4 - {x^2})\) , for \( - 2 < x < 2\) . Consider \(g'(x) = w\) ....
• SPNone.1.sl.TZ0.7a: Find the first four derivatives of \(f(x)\) .
• SPNone.1.sl.TZ0.7b: Write an expression for \({f^{(n)}}(x)\) in terms of x and n .
• SPNone.1.sl.TZ0.10a: Find \(f'(x)\) .
• 11N.1.sl.TZ0.9a(i), (ii) and (iii): Use the graph to write down the value of (i)     a ; (ii)    c ; (iii)   d .
• 11N.1.sl.TZ0.9b: Show that \(b = \frac{\pi }{4}\) .
• 11N.1.sl.TZ0.9c: Find \(f'(x)\) .
• 11N.1.sl.TZ0.9d: At a point R, the gradient is \( - 2\pi \) . Find the x-coordinate of R.
• 11N.2.sl.TZ0.10a: Sketch the graph of f .
• 11N.2.sl.TZ0.10b(i) and (ii): (i)     Write down the x-coordinate of the maximum point on the graph of f . (ii)    Write down...
• 11N.2.sl.TZ0.10c: Show that \(f'(x) = \frac{{20 - 6x}}{{{{\rm{e}}^{0.3x}}}}\) .
• 11N.2.sl.TZ0.10d: Find the interval where the rate of change of f is increasing.
• 11M.1.sl.TZ1.5a: Use the quotient rule to show that \(g'(x) = \frac{{1 - 2\ln x}}{{{x^3}}}\) .
• 11M.1.sl.TZ1.5b: The graph of g has a maximum point at A. Find the x-coordinate of A.
• 11M.2.sl.TZ1.9a: Identify the two points of inflexion.
• 11M.2.sl.TZ1.9b(i) and (ii): (i)     Find \(f'(x)\) . (ii)    Show that \(f''(x) = (4{x^2} - 2){{\rm{e}}^{ - {x^2}}}\) .
• 11M.2.sl.TZ1.9c: Find the x-coordinate of each point of inflexion.
• 11M.2.sl.TZ1.9d: Use the second derivative to show that one of these points is a point of inflexion.
• 11M.1.sl.TZ2.4: Let \(h(x) = \frac{{6x}}{{\cos x}}\) . Find \(h'(0)\) .
• 11M.1.sl.TZ2.9a: Write down \(f(x)\) in the form \(f(x) = - 10(x - p)(x - q)\) .
• 11M.1.sl.TZ2.9b: Find another expression for \(f(x)\) in the form \(f(x) = - 10{(x - h)^2} + k\) .
• 11M.1.sl.TZ2.9c: Show that \(f(x)\) can also be written in the form \(f(x) = 240 + 20x - 10{x^2}\) .
• 11M.1.sl.TZ2.9d(i) and (ii): A particle moves along a straight line so that its velocity, \(v{\text{ m}}{{\text{s}}^{ - 1}}\)...
• 13M.1.sl.TZ1.3a: Find \(f'(x)\) .
• 13M.1.sl.TZ1.10c: (i)     Find \(f''(1)\) . (ii)     Hence, show that there is no point of inflexion on the graph...
• 13M.2.sl.TZ1.9d: Show that \(f'(x) = \frac{{1000{{\rm{e}}^{ - 0.2x}}}}{{{{(1 + 50{{\rm{e}}^{ - 0.2x}})}^2}}}\) .
• 13M.1.sl.TZ2.9a: Find \(f'(x)\) .
• 13M.1.sl.TZ2.10d: find the equation of the normal to the graph of \(h\) at P.
• 13M.2.sl.TZ2.10b: Consider all values of \(m\) such that the graphs of \(f\) and \(g\) intersect. Find the value of...
• 14M.1.sl.TZ1.7a: Find \(f'(x)\).
• 14M.2.sl.TZ1.7: Let \(f(x) = \frac{{g(x)}}{{h(x)}}\), where \(g(2) = 18,{\text{ }}h(2) = 6,{\text{ }}g'(2) = 5\),...
• 14M.1.sl.TZ2.10a: Use the quotient rule to show that \(f'(x) = \frac{{10 - 2{x^2}}}{{{{({x^2} + 5)}^2}}}\).
• 13N.1.sl.TZ0.10a: Show that \(f'(x) = \frac{{\ln x}}{x}\).
• 13N.2.sl.TZ0.3a: Find \(f'(x)\).
• 15M.1.sl.TZ1.9a: Find \(f''(x)\).
• 15M.1.sl.TZ1.9c: Find \(f'( - 2)\).
• 15N.1.sl.TZ0.10d: The following diagram shows the shaded regions \(A\), \(B\) and \(C\). The regions are...
• 16M.1.sl.TZ1.9a: Find the \(x\)-coordinate of P.
• 16M.1.sl.TZ1.9b: Find \(f(x)\), expressing your answer as a single logarithm.
• 16M.1.sl.TZ1.9c: The graph of \(f\) is transformed by a vertical stretch with scale factor \(\frac{1}{{\ln 3}}\)....
• 16M.1.sl.TZ1.10a: Find \(f'(1)\).
• 16M.1.sl.TZ1.10b: Write down \(g'(1)\).
• 16M.1.sl.TZ1.10c: Find \(g(1)\).
• 16M.1.sl.TZ1.10d: Let \(h(x) = f(x) \times g(x)\). Find the equation of the tangent to the graph of \(h\) at the...
• 16M.1.sl.TZ2.9a: Show that \(A(x) = \frac{{108}}{x} + 2{x^2}\).
• 16M.1.sl.TZ2.9b: Find \(A'(x)\).
• 16M.1.sl.TZ2.9c: Given that the outside surface area is a minimum, find the height of the container.
• 16M.1.sl.TZ2.9d: Fred paints the outside of the container. A tin of paint covers a surface area of...
• 16M.2.sl.TZ2.9a: Write down the equation of the horizontal asymptote of the graph of \(f\).
• 16M.2.sl.TZ2.9b: Find \(f'(x)\).
• 16M.2.sl.TZ2.9c: Write down the value of \(b\).
• 16M.2.sl.TZ2.9d: Given that \(g'(1) =  - e\), find the value of \(a\).
• 16M.2.sl.TZ2.9e: There is a value of \(x\), for \(1 < x < 4\), for which the graphs of \(f\) and \(g\) have...
• 16N.1.sl.TZ0.10a: (i)     Find the first four derivatives of \(f(x)\). (ii)     Find \({f^{(19)}}(x)\).
• 16N.1.sl.TZ0.10b: (i)     Find the first three derivatives of \(g(x)\). (ii)     Given that...
• 16N.1.sl.TZ0.10c: (i)     Find \(h'(x)\). (ii)     Hence, show that \(h'(\pi ) = \frac{{ - 21!}}{2}{\pi ^2}\).
• 17M.1.sl.TZ2.6a: Find \(h(1)\).
• 17M.1.sl.TZ2.6b: Find \(h'(8)\).
• 17M.1.sl.TZ2.10a.i: Write down \(f'(x)\).
• 17M.1.sl.TZ2.10a.ii: Find the gradient of \(L\).
• 17M.1.sl.TZ2.10b: Show that the \(x\)-coordinate of B is \( - \frac{k}{2}\).
• 17M.1.sl.TZ2.10c: Find the area of triangle ABC, giving your answer in terms of \(k\).
• 17M.1.sl.TZ2.10d: Given that the area of triangle ABC is \(p\) times the area of \(R\), find the value of \(p\).
• 17M.2.sl.TZ1.6: Let \(f(x) = {({x^2} + 3)^7}\). Find the term in \({x^5}\) in the expansion of the derivative,...
• 17N.1.sl.TZ0.8a: Show that \(f’(1) = 1\).
• 17N.1.sl.TZ0.8b: Find the equation of \(L\) in the form \(y = ax + b\).
• 17N.1.sl.TZ0.8d: Find the area of the region enclosed by the graph of \(f\) and the line \(L\).
• 17N.1.sl.TZ0.8c: Find the \(x\)-coordinate of Q.
• 18M.1.sl.TZ1.7: Consider f(x), g(x) and h(x), for x∈\(\mathbb{R}\) where h(x) = \(\left( {f \circ g}...
• 18M.1.sl.TZ1.8a: Find f (x).
• 18M.1.sl.TZ1.8b: The graph of f has a point of inflexion at x = p. Find p.
• 18M.1.sl.TZ1.8c: Find the values of x for which the graph of f is concave-down.
• 18M.2.sl.TZ1.1a: Find f '(x).
• 18M.2.sl.TZ1.1b: Find f "(x).
• 18M.2.sl.TZ1.1c: Solve f '(x) = f "(x).
• 18M.1.sl.TZ2.9a: Express h in terms of r.
• 18M.1.sl.TZ2.9b: Show that \(C = 20\pi {r^2} + \frac{{320\pi }}{r}\).
• 18M.1.sl.TZ2.9c: Given that there is a minimum value for C, find this minimum value in terms of \(\pi \).
• 18M.1.sl.TZ2.10a.i: Write down \(f'\left( 2 \right)\).
• 18M.1.sl.TZ2.10a.ii: Find \(f\left( 2 \right)\).
• 18M.1.sl.TZ2.10b: Show that the graph of g has a gradient of 6 at P.
• 18M.1.sl.TZ2.10c: Let L2 be the tangent to the graph of g at P. L1 intersects L2 at the point Q. Find the...

6.3

• 12M.1.sl.TZ2.10a: Use the quotient rule to show that \(f'(x) = \frac{{2{x^2} - 2}}{{{{( - 2{x^2} + 5x - 2)}^2}}}\) .
• 12M.1.sl.TZ2.10b: Hence find the coordinates of B.
• 12M.1.sl.TZ2.10c: Given that the line \(y = k\) does not meet the graph of f , find the possible values of k .
• 08N.1.sl.TZ0.6a(i) and (ii): (i)     Write down the value of \(f'(x)\) at C. (ii)    Hence, show that C corresponds to a...
• 08N.1.sl.TZ0.6b: Which of the points A, B, D corresponds to a maximum on the graph of f ?
• 08N.1.sl.TZ0.6c: Show that B corresponds to a point of inflexion on the graph of f .
• 08M.1.sl.TZ1.8b: Find the x-coordinate of M.
• 08M.1.sl.TZ1.8c: Find the x-coordinate of N.
• 08M.2.sl.TZ1.5a: On the grid below, sketch a graph of \(y = f''(x)\) , clearly indicating the x-intercept.
• 08M.2.sl.TZ1.5b: Complete the table, for the graph of \(y = f(x)\) .
• 08M.2.sl.TZ1.5c: Justify your answer to part (b) (ii).
• 08M.1.sl.TZ2.10d: Find the value of \(\theta \) when S is a local minimum, justifying that it is a minimum.
• 08M.1.sl.TZ2.10e: Find a value of \(\theta \) for which S has its greatest value.
• 08M.2.sl.TZ2.9c: Write down the value of r and of s.
• 10M.1.sl.TZ1.8a: Find the coordinates of A.
• 10M.1.sl.TZ1.8b(i), (ii) and (iii): Write down the coordinates of (i)     the image of B after reflection in the y-axis; (ii)   ...
• 10M.1.sl.TZ1.9a: Use the quotient rule to show that \(f'(x) = \frac{{ - 1}}{{{{\sin }^2}x}}\) .
• 10M.1.sl.TZ1.9b: Find \(f''(x)\) .
• 10M.1.sl.TZ1.9c: Find the value of p and of q.
• 10M.1.sl.TZ1.9d: Use information from the table to explain why there is a point of inflexion on the graph of f...
• 10M.1.sl.TZ2.7a: Write down the x-intercepts of the graph of the derivative function, \(f'\) .
• 10M.1.sl.TZ2.7b: Write down all values of x for which \(f'(x)\) is positive.
• 10M.1.sl.TZ2.7c: At point D on the graph of f , the x-coordinate is \( - 0.5\). Explain why \(f''(x) < 0\) at D.
• 10M.1.sl.TZ2.8a: Use the second derivative to justify that B is a maximum.
• 10M.1.sl.TZ2.8b: Given that \(f'(x) = \frac{3}{2}{x^2} - x + p\) , show that \(p = - 4\) .
• 10M.1.sl.TZ2.8c: Find \(f(x)\) .
• 09N.1.sl.TZ0.5b: There is a minimum value of \(f(x)\) when \(x = - 2\) . Find the value of \(p\) .
• 09N.1.sl.TZ0.9b: The second derivative \(f''(x) = \frac{{40(3{x^2} + 4)}}{{{{({x^2} - 4)}^3}}}\) . Use this...
• 09N.1.sl.TZ0.9c: Describe the behaviour of the graph of \(f\) for large \(|x|\) .
• 09N.2.sl.TZ0.7: The fencing used for side AB costs \(\$ 11\) per metre. The fencing for the other three sides...
• 09M.1.sl.TZ1.4a: Complete the following table by noting which graph A, B or C corresponds to each function.
• 09M.1.sl.TZ1.4b: Write down the value of t when the velocity is greatest.
• 09M.1.sl.TZ1.8c: (i)     Find \(\frac{{{\rm{d}}A}}{{{\rm{d}}\theta }}\) . (ii)    Hence, find the exact value of...
• 09M.1.sl.TZ1.10b: Given that \(f''(x) = \frac{{2ax({x^2} - 3)}}{{{{({x^2} + 1)}^3}}}\) , find the coordinates of...
• 09M.1.sl.TZ2.6c: The point P on the graph of f has x-coordinate \(3\). Explain why P is not a point of inflexion.
• 10N.2.sl.TZ0.7a: There are two points of inflexion on the graph of f . Write down the x-coordinates of these points.
• 10N.2.sl.TZ0.7b: Let \(g(x) = f''(x)\) . Explain why the graph of g has no points of inflexion.
• SPNone.1.sl.TZ0.10b(i) and (ii): Hence (i)     show that \(q = - 2\) ; (ii)    verify that A is a minimum point.
• 11M.1.sl.TZ1.5a: Use the quotient rule to show that \(g'(x) = \frac{{1 - 2\ln x}}{{{x^3}}}\) .
• 11M.1.sl.TZ1.5b: The graph of g has a maximum point at A. Find the x-coordinate of A.
• 11M.2.sl.TZ1.9a: Identify the two points of inflexion.
• 11M.2.sl.TZ1.9b(i) and (ii): (i)     Find \(f'(x)\) . (ii)    Show that \(f''(x) = (4{x^2} - 2){{\rm{e}}^{ - {x^2}}}\) .
• 11M.2.sl.TZ1.9c: Find the x-coordinate of each point of inflexion.
• 11M.2.sl.TZ1.9d: Use the second derivative to show that one of these points is a point of inflexion.
• 11M.2.sl.TZ2.10a: Show that the area of the window is given by \(y = 4\sin \theta + 2\sin 2\theta \) .
• 11M.2.sl.TZ2.10b: Zoe wants a window to have an area of \(5{\text{ }}{{\text{m}}^2}\). Find the two possible values...
• 11M.2.sl.TZ2.10c: John wants two windows which have the same area A but different values of \(\theta \) . Find all...
• 13M.1.sl.TZ1.10c: (i)     Find \(f''(1)\) . (ii)     Hence, show that there is no point of inflexion on the graph...
• 13M.1.sl.TZ1.10d: There is a point of inflexion on the graph of \(f\) at \(x = \sqrt[4]{3}\)...
• 13M.2.sl.TZ1.9e: Find the maximum rate of change of \(f\) .
• 13M.1.sl.TZ2.10b: Explain why P is a point of inflexion.
• 14M.2.sl.TZ1.10c: The vertical and horizontal asymptotes to the graph of \(f\) intersect at the...
• 14M.2.sl.TZ1.10d: The vertical and horizontal asymptotes to the graph of \(f\) intersect at the...
• 14M.1.sl.TZ2.6a: On the following axes, sketch the graph of \(y = f'(x)\).
• 14M.1.sl.TZ2.6b: Write down the following in order from least to greatest:...
• 13N.1.sl.TZ0.10b: There is a minimum on the graph of \(f\). Find the \(x\)-coordinate of this minimum.
• 13N.1.sl.TZ0.10c: Write down the value of \(p\).
• 13N.2.sl.TZ0.7a: Let \({\text{OP}} = x\). (i)     Find \({\text{PQ}}\), giving your answer in terms of...
• 14N.1.sl.TZ0.9a: Find the \(x\)-coordinate of \(A\).
• 14N.1.sl.TZ0.10d: There is a minimum value for \(d\). Find the value of \(a\) that gives this minimum value.
• 15M.1.sl.TZ1.9b: The graph of \(f\) has a point of inflexion when \(x = 1\). Show that \(k = 3\).
• 15M.1.sl.TZ1.9e: Given that \(f'( - 1) = 0\), explain why the graph of \(f\) has a local maximum when \(x =  - 1\).
• 15M.2.sl.TZ1.10b: Write down \(f'(2)\).
• 15N.1.sl.TZ0.10a: Explain why the graph of \(f\) has a local minimum when \(x = 5\).
• 15N.1.sl.TZ0.10b: Find the set of values of \(x\) for which the graph of \(f\) is concave down.
• 16M.1.sl.TZ1.9a: Find the \(x\)-coordinate of P.
• 16M.1.sl.TZ1.9b: Find \(f(x)\), expressing your answer as a single logarithm.
• 16M.1.sl.TZ1.9c: The graph of \(f\) is transformed by a vertical stretch with scale factor \(\frac{1}{{\ln 3}}\)....
• 16M.1.sl.TZ2.9a: Show that \(A(x) = \frac{{108}}{x} + 2{x^2}\).
• 16M.1.sl.TZ2.9b: Find \(A'(x)\).
• 16M.1.sl.TZ2.9c: Given that the outside surface area is a minimum, find the height of the container.
• 16M.1.sl.TZ2.9d: Fred paints the outside of the container. A tin of paint covers a surface area of...
• 16N.2.sl.TZ0.10a: (i)     Find the value of \(c\). (ii)     Show that \(b = \frac{\pi }{6}\). (iii)     Find the...
• 16N.2.sl.TZ0.10b: (i)     Write down the value of \(k\). (ii)     Find \(g(x)\).
• 16N.2.sl.TZ0.10c: (i)     Find \(w\). (ii)     Hence or otherwise, find the maximum positive rate of change of \(g\).
• 17M.1.sl.TZ1.6a.i: Write down the gradient of the curve of \(f\) at P.
• 17M.1.sl.TZ1.6a.ii: Find the equation of the normal to the curve of \(f\) at P.
• 17M.1.sl.TZ1.6b: Determine the concavity of the graph of \(f\) when \(4 < x < 5\) and justify your answer.
• 17M.2.sl.TZ1.10a.i: Write down the value of \(q\);
• 17M.2.sl.TZ1.10a.ii: Write down the value of \(h\);
• 17M.2.sl.TZ1.10a.iii: Write down the value of \(k\).
• 17M.2.sl.TZ1.10b.i: Find \(\int_{0.111}^{3.31} {\left( {h(x) - x} \right){\text{d}}x} \).
• 17M.2.sl.TZ1.10b.ii: Hence, find the area of the region enclosed by the graphs of \(h\) and \({h^{ - 1}}\).
• 17M.2.sl.TZ1.10c: Let \(d\) be the vertical distance from a point on the graph of \(h\) to the line \(y = x\)....
• 17M.2.sl.TZ2.8a: Find the value of \(p\).
• 17M.2.sl.TZ2.8b.i: Write down the coordinates of A.
• 17M.2.sl.TZ2.8b.ii: Write down the rate of change of \(f\) at A.
• 17M.2.sl.TZ2.8c.i: Find the coordinates of B.
• 17M.2.sl.TZ2.8c.ii: Find the the rate of change of \(f\) at B.
• 17M.2.sl.TZ2.8d: Let \(R\) be the region enclosed by the graph of \(f\) , the \(x\)-axis, the line \(x = b\) and...
• 17N.1.sl.TZ0.7: Consider \(f(x) = \log k(6x - 3{x^2})\), for \(0 < x < 2\), where \(k > 0\). The...
• 18M.1.sl.TZ1.8a: Find f (x).
• 18M.1.sl.TZ1.8b: The graph of f has a point of inflexion at x = p. Find p.
• 18M.1.sl.TZ1.8c: Find the values of x for which the graph of f is concave-down.
• 18M.2.sl.TZ1.10a: Find the coordinates of A.
• 18M.2.sl.TZ1.10b.i: For the graph of \(f\), write down the amplitude.
• 18M.2.sl.TZ1.10c: Hence, write \(f\left( x \right)\) in the form \(p\,\,{\text{cos}}\,\left( {x + r} \right)\).
• 18M.2.sl.TZ1.10d: Find the maximum speed of the ball.
• 18M.2.sl.TZ1.10e: Find the first time when the ball’s speed is changing at a rate of 2 cm s−2.
• 18M.2.sl.TZ1.10b.ii: For the graph of \(f\), write down the period.
• 18M.1.sl.TZ2.9a: Express h in terms of r.
• 18M.1.sl.TZ2.9b: Show that \(C = 20\pi {r^2} + \frac{{320\pi }}{r}\).
• 18M.1.sl.TZ2.9c: Given that there is a minimum value for C, find this minimum value in terms of \(\pi \).
• 18M.2.sl.TZ2.9a: Find the initial velocity of P.
• 18M.2.sl.TZ2.9b: Find the maximum speed of P.
• 18M.2.sl.TZ2.9c: Write down the number of times that the acceleration of P is 0 m s−2 .
• 18M.2.sl.TZ2.9d: Find the acceleration of P when it changes direction.
• 18M.2.sl.TZ2.9e: Find the total distance travelled by P.

6.4

• 08M.1.sl.TZ1.5a: Find  \(\int {\frac{1}{{2x + 3}}} {\rm{d}}x\) .
• 08M.1.sl.TZ2.9c: Let \(g(x) = \sqrt 3 \sin x{(\cos x)^{\frac{1}{2}}}\) for \(0 \le x \le \frac{\pi }{2}\) . Find...
• 12M.1.sl.TZ1.6: Given that \(\int_0^5 {\frac{2}{{2x + 5}}} {\rm{d}}x = \ln k\) , find the value of k .
• SPNone.1.sl.TZ0.5a: Find \(\int {\frac{{{{\rm{e}}^x}}}{{1 + {{\rm{e}}^x}}}} {\rm{d}}x\) .
• SPNone.1.sl.TZ0.5b: Find \(\int {\sin 3x\cos 3x{\rm{d}}x} \) .
• 13M.1.sl.TZ1.6: Let \(f(x) = \int {\frac{{12}}{{2x - 5}}} {\rm{d}}x\) , \(x > \frac{5}{2}\) . The graph of...
• 14M.1.sl.TZ2.10b: Find \(\int {\frac{{2x}}{{{x^2} + 5}}{\text{d}}x} \).
• 13N.2.sl.TZ0.3b: Find \(\int {f(x){\text{d}}x} \).
• 14N.1.sl.TZ0.6: The following diagram shows the graph of \(f(x) = \frac{x}{{{x^2} + 1}}\), for \(0 \le x \le 4\),...
• 15M.2.sl.TZ1.10d: Verify that \(\ln 3 + \int_2^a {g'(x){\text{d}}x = g(a)} \), where \(0 \le a \le 10\).
• 16M.1.sl.TZ1.9a: Find the \(x\)-coordinate of P.
• 16M.1.sl.TZ1.9b: Find \(f(x)\), expressing your answer as a single logarithm.
• 16M.1.sl.TZ1.9c: The graph of \(f\) is transformed by a vertical stretch with scale factor \(\frac{1}{{\ln 3}}\)....
• 16N.1.sl.TZ0.6: Let \(f'(x) = {\sin ^3}(2x)\cos (2x)\). Find \(f(x)\), given that...
• 17M.1.sl.TZ1.5a: Find \(\int {x{{\text{e}}^{{x^2} - 1}}{\text{d}}x} \).
• 17M.1.sl.TZ1.5b: Find \(f(x)\), given that \(f’(x) = x{{\text{e}}^{{x^2} - 1}}\) and \(f( - 1) = 3\).
• 17M.1.sl.TZ2.5: Let \(f’(x) = \frac{{3{x^2}}}{{{{({x^3} + 1)}^5}}}\). Given that \(f(0) = 1\), find \(f(x)\).
• 17M.2.sl.TZ1.7a.i: Write down the first value of \(t\) at which P changes direction.
• 17M.2.sl.TZ1.7a.ii: Find the total distance travelled by P, for \(0 \leqslant t \leqslant 8\).
• 17M.2.sl.TZ1.7b: A second particle Q also moves along a straight line. Its velocity,...
• 18M.1.sl.TZ1.5a: Find \(\int {{{\left( {f\left( x \right)} \right)}^2}{\text{d}}x} \).
• 18M.1.sl.TZ1.5b: Part of the graph of f is shown in the following diagram. The shaded region R is enclosed by...
• 18M.1.sl.TZ1.8a: Find f (x).
• 18M.1.sl.TZ1.8b: The graph of f has a point of inflexion at x = p. Find p.
• 18M.1.sl.TZ1.8c: Find the values of x for which the graph of f is concave-down.
• 18M.1.sl.TZ2.2a: Find \(\int {\left( {6{x^2} - 3x} \right){\text{d}}x} \).
• 18M.1.sl.TZ2.2b: Find the area of the region enclosed by the graph of \(f\), the x-axis and the lines x = 1 and x...

6.5

• 12N.1.sl.TZ0.3a: Find \(\int_4^{10} {(x - 4){\rm{d}}x} \) .
• 12N.1.sl.TZ0.3b: Part of the graph of \(f(x) = \sqrt {{x^{}} - 4} \) , for \(x \ge 4\) , is shown below. The...
• 12N.1.sl.TZ0.10a: Find \(f'(x)\) .
• 12N.2.sl.TZ0.9a: Sketch the graph of f , for \( - 1 \le x \le 5\) .
• 12N.2.sl.TZ0.9b: This function can also be written as \(f(x) = {(x - p)^2} - 3\) . Write down the value of p .
• 12N.2.sl.TZ0.9c: The graph of g is obtained by reflecting the graph of f in the x-axis, followed by a translation...
• 12N.2.sl.TZ0.9d: The graph of g is obtained by reflecting the graph of f in the x-axis, followed by a translation...
• 12N.2.sl.TZ0.9e: The graph of \(g\) is obtained by reflecting the graph of \(f\) in the x-axis, followed by a...
• 12N.1.sl.TZ0.10b: Let \(g(x) = \ln \left( {\frac{{6x}}{{x + 1}}} \right)\) , for \(x > 0\) . Show that...
• 12M.1.sl.TZ2.8a(i) and (ii): The function can be written in the form \(f(x) = a{(x - h)^2} + k\) . (i)     Write down the...
• 12M.1.sl.TZ2.8b: Find \(f(x)\)  , giving your answer in the form \(A{x^2} + Bx + C\) .
• 12M.1.sl.TZ2.8c: Calculate the area enclosed by the graph of f , the x-axis, and the lines \(x = 2\) and \(x = 4\) .
• 12N.1.sl.TZ0.10c: Let \(h(x) = \frac{1}{{x(x + 1)}}\) . The area enclosed by the graph of h , the x-axis and the...
• 08N.2.sl.TZ0.4c: The graph of f is revolved \(360^\circ \) about the x-axis from \(x = 0\) to  \(x = a\) . Find...
• 08N.2.sl.TZ0.9c(i) and (ii): The graph of f and the line L intersect at the point (0, 1) and at a second point P. (i)    ...
• 08M.1.sl.TZ1.5b: Given that \(\int_0^3 {\frac{1}{{2x + 3}}} {\rm{d}}x = \ln \sqrt P \) , find the value of P.
• 08M.2.sl.TZ1.10a(i) and (ii): Let A be the area of the region enclosed by the curves of f and g. (i)     Find an expression...
• 08M.1.sl.TZ2.7a: Show that \(\int_5^1 {f(x){\rm{d}}x = - 4} \) .
• 08M.1.sl.TZ2.7b: Find the value of  \(\int_1^2 {(x + f(x)){\rm{d}}x + } \int_2^5 {(x + f(x)){\rm{d}}x} \) .
• 08M.1.sl.TZ2.9c: Let \(g(x) = \sqrt 3 \sin x{(\cos x)^{\frac{1}{2}}}\) for \(0 \le x \le \frac{\pi }{2}\) . Find...
• 08M.2.sl.TZ2.9e(i) and (ii): Let R be the region enclosed by the curve \(y = f(x)\) and the line L. (i)     Find an...
• 12M.1.sl.TZ1.6: Given that \(\int_0^5 {\frac{2}{{2x + 5}}} {\rm{d}}x = \ln k\) , find the value of k .
• 12M.1.sl.TZ1.10a: Find \(s'(t)\) .
• 12M.1.sl.TZ1.10b: In this interval, there are only two values of t for which the object is not moving. One value is...
• 12M.1.sl.TZ1.10c: Show that \(s'(t) > 0\) between these two values of t .
• 12M.1.sl.TZ1.10d: Find the distance travelled between these two values of t .
• 12M.2.sl.TZ1.4a: Find k .
• 12M.2.sl.TZ1.4b: The shaded region is rotated \(360^\circ \) about the x-axis. Let V be the volume of the solid...
• 12M.2.sl.TZ1.4c: The shaded region is rotated \(360^\circ \) about the x-axis. Let V be the volume of the solid...
• 10N.1.sl.TZ0.6: The graph of the function \(y = f(x)\) passes through the point...
• 10N.1.sl.TZ0.10a(i), (ii) and (iii): (i)     Show that the gradient of [PQ] is \(\frac{{{a^3}}}{{a - \frac{2}{3}}}\) . (ii)    Find...
• 10N.1.sl.TZ0.10b: Given that the area of T is \(2k + 4\) , show that k satisfies the equation...
• 10M.1.sl.TZ1.6: The region enclosed by the curve of f and the x-axis is rotated \(360^\circ \) about the...
• 10M.1.sl.TZ2.8a: Use the second derivative to justify that B is a maximum.
• 10M.1.sl.TZ2.8b: Given that \(f'(x) = \frac{3}{2}{x^2} - x + p\) , show that \(p = - 4\) .
• 10M.1.sl.TZ2.8c: Find \(f(x)\) .
• 10M.1.sl.TZ2.10a(i) and (ii): Solve for \(0 \le x < 2\pi \) (i)     \(6 + 6\sin x = 6\) ; (ii)    \(6 + 6\sin x = 0\) .
• 10M.1.sl.TZ2.10b: Write down the exact value of the x-intercept of f , for \(0 \le x < 2\pi \) .
• 10M.1.sl.TZ2.10c: The area of the shaded region is k . Find the value of k , giving your answer in terms of \(\pi \) .
• 10M.1.sl.TZ2.10d: Let \(g(x) = 6 + 6\sin \left( {x - \frac{\pi }{2}} \right)\) . The graph of f is transformed to...
• 10M.1.sl.TZ2.10e: Let \(g(x) = 6 + 6\sin \left( {x - \frac{\pi }{2}} \right)\) . The graph of f is transformed to...
• 09N.1.sl.TZ0.10c: Find an expression for the area of R .
• 09N.1.sl.TZ0.10d: The region R is rotated \(360^\circ \) about the x-axis. Find the volume of the solid formed,...
• 09N.2.sl.TZ0.9d: Let R be the region enclosed by the graphs of f and g . Find the area of R.
• 09M.1.sl.TZ1.7: The graph of \(y = \sqrt x \) between \(x = 0\) and \(x = a\) is rotated \(360^\circ \) about the...
• 09M.1.sl.TZ1.10c: It is given that \(\int {f(x){\rm{d}}x = \frac{a}{2}} \ln ({x^2} + 1) + C\) . (i)     Find the...
• 09M.2.sl.TZ2.8a: Find the area of R.
• 09M.2.sl.TZ2.8b: Find the volume of the solid formed when R is rotated through \({360^ \circ }\) about the x-axis.
• 09M.2.sl.TZ2.8c: The diagram below shows a part of the graph of a quadratic function \(g(x) = x(a - x)\) . The...
• 10N.2.sl.TZ0.2a: On the grid below, sketch the graph of v , clearly indicating the maximum point.
• 10N.2.sl.TZ0.2b(i) and (ii): (i)     Write down an expression for d . (ii)    Hence, write down the value of d .
• 10N.2.sl.TZ0.8a: Find the value of a and of b .
• 10N.2.sl.TZ0.8b: The graph of f has a maximum value when \(x = c\) . Find the value of c .
• 10N.2.sl.TZ0.8c: The region under the graph of f from \(x = 0\) to \(x = c\) is rotated \({360^ \circ }\) about...
• 10N.2.sl.TZ0.8d: Let R be the region enclosed by the curve, the x-axis and the line \(x = c\) , between \(x = a\)...
• 10M.2.sl.TZ1.6: The acceleration,  \(a{\text{ m}}{{\text{s}}^{ - 2}}\), of a particle at time t seconds is given...
• 10M.2.sl.TZ1.9a: Show that \(A = 10\) .
• 10M.2.sl.TZ1.9b: Given that \(f(15) = 3.49\) (correct to 3 significant figures), find the value of k.
• 10M.2.sl.TZ1.9c(i), (ii) and (iii): (i)     Using your value of k , find \(f'(x)\) . (ii)    Hence, explain why f is a decreasing...
• 10M.2.sl.TZ1.9d: Let \(g(x) = - {x^2} + 12x - 24\) . Find the area enclosed by the graphs of f and g .
• 10M.2.sl.TZ2.6a: Write down the x-coordinate of A.
• 10M.2.sl.TZ2.6b(i) and (ii): Find the value of (i)     p ; (ii)    q .
• 10M.2.sl.TZ2.6c: Find \(\int_p^q {f(x){\rm{d}}x} \) . Explain why this is not the area of the shaded region.
• SPNone.2.sl.TZ0.9c(i) and (ii): Let R be the region in the first quadrant enclosed by the graph of h , the x-axis and the line...
• 11N.1.sl.TZ0.4: Let \(f'(x) = 3{x^2} + 2\) . Given that \(f(2) = 5\) , find \(f(x)\) .
• 11N.1.sl.TZ0.10a: Find the equation of L .
• 11N.1.sl.TZ0.10b: Find the area of the region enclosed by the curve of g , the x-axis, and the lines \(x = 2\) and...
• 11N.1.sl.TZ0.10c: The graph of g is reflected in the x-axis to give the graph of h . The area of the region...
• 11M.2.sl.TZ1.6: Let \(f(x) = \cos ({x^2})\) and \(g(x) = {{\rm{e}}^x}\) , for \( - 1.5 \le x \le 0.5\) . Find...
• 11M.1.sl.TZ2.8a: Show that the equation of T is \(y = 4x - 2\) .
• 11M.1.sl.TZ2.8b: Find the x-intercept of T .
• 11M.1.sl.TZ2.8c(i) and (ii): The shaded region R is enclosed by the graph of f , the line T , and the x-axis. (i)     Write...
• 11M.2.sl.TZ2.7: A gradient function is given by \(\frac{{{\rm{d}}y}}{{{\rm{d}}x}} = 10{{\rm{e}}^{2x}} - 5\) ....
• 13M.1.sl.TZ1.6: Let \(f(x) = \int {\frac{{12}}{{2x - 5}}} {\rm{d}}x\) , \(x > \frac{5}{2}\) . The graph of...
• 13M.1.sl.TZ2.6: A rocket moving in a straight line has velocity \(v\) km s–1 and displacement \(s\) km at time...
• 14M.1.sl.TZ1.3a: Find \(\int_1^2 {{{\left( {f(x)} \right)}^2}{\text{d}}x} \).
• 14M.1.sl.TZ1.3b: The following diagram shows part of the graph of \(f\).     The shaded region \(R\) is...
• 14M.1.sl.TZ1.6: Let \(\int_\pi ^a {\cos 2x{\text{d}}x}  = \frac{1}{2}{\text{, where }}\pi  < a < 2\pi \)....
• 14M.1.sl.TZ2.5: The graph of a function h passes through the point \(\left( {\frac{\pi }{{12}}, 5}...
• 14M.1.sl.TZ2.10c: The following diagram shows part of the graph of \(f\). The shaded region is enclosed by the...
• 14M.2.sl.TZ2.2b: The region enclosed by the graph of \(f\) and the \(x\)-axis is revolved \(360^\circ \) about the...
• 13N.1.sl.TZ0.4a: Find \(\int_1^6 {2f(x){\text{d}}x} \).
• 13N.1.sl.TZ0.4b: Find \(\int_1^6 {\left( {f(x) + 2} \right){\text{d}}x} \).
• 13N.1.sl.TZ0.10e: The graph of \(g\) intersects the graph of \(f'\) when \(x = q\). Let \(R\) be the region...
• 13N.2.sl.TZ0.2b: The region enclosed by the graph of \(f\) and the \(x\)-axis is rotated \(360^\circ\) about the...
• 13M.1.sl.TZ2.7a: Find \(\int_0^2 {f(x){\rm{d}}x} \) .
• 13M.1.sl.TZ2.7b: The shaded region is enclosed by the graph of \(f\) , the \(x\)-axis, the \(y\)-axis and the line...
• 14N.1.sl.TZ0.6: The following diagram shows the graph of \(f(x) = \frac{x}{{{x^2} + 1}}\), for \(0 \le x \le 4\),...
• 14N.1.sl.TZ0.9b: The \(y\)-intercept of the graph is at (\(0,6\)). Find an expression for \(f(x)\). The graph of...
• 14N.2.sl.TZ0.4c: The region enclosed by the graph of \(f\) and the \(x\)-axis is rotated \(360°\) about the...
• 15M.1.sl.TZ1.7: Let \(f(x) = \cos x\), for \(0\) \(\le \) \(x\) \( \le \) \(2\pi \). The following diagram shows...
• 15M.2.sl.TZ1.10d: Verify that \(\ln 3 + \int_2^a {g'(x){\text{d}}x = g(a)} \), where \(0 \le a \le 10\).
• 15M.2.sl.TZ1.10e: The following diagram shows the graph of \(g'\), the derivative of \(g\). The shaded region...
• 15N.1.sl.TZ0.3: Let \(f'(x) = 6{x^2} - 5\). Given that \(f(2) =  - 3\), find \(f(x)\).
• 15N.1.sl.TZ0.10c: The following diagram shows the shaded regions \(A\), \(B\) and \(C\). The regions are...
• 15N.2.sl.TZ0.3c: The region enclosed by the graph of \(f\), the \(x\)-axis and the line \(x = 10\) is rotated...
• 16M.2.sl.TZ1.2a: Solve \(f(x) = g(x)\).
• 16M.2.sl.TZ1.2b: Find the area of the region enclosed by the graphs of \(f\) and \(g\).
• 16M.2.sl.TZ1.9a: Find the displacement of P from O after 5 seconds.
• 16M.2.sl.TZ1.9b: Find when P is first at rest.
• 16M.2.sl.TZ1.9c: Write down the number of times P changes direction.
• 16M.2.sl.TZ1.9d: Find the acceleration of P after 3 seconds.
• 16M.2.sl.TZ1.9e: Find the maximum speed of P.
• 16M.1.sl.TZ2.10a: (i)     Given that \(f'(x) = \frac{{2{a^2} - 4{x^2}}}{{\sqrt {{a^2} - {x^2}} }}\), for...
• 16M.1.sl.TZ2.10b: Show that \({A_R} = \frac{2}{3}{a^3}\).
• 16M.1.sl.TZ2.10c: Let \({A_T}\) be the area of the triangle OPQ. Given that \({A_T} = k{A_R}\), find the value of...
• 16M.2.sl.TZ2.7: A particle moves in a straight line. Its velocity \(v{\text{ m}}\,{{\text{s}}^{ - 1}}\) after...
• 16N.2.sl.TZ0.4b: Hence, find the area of the region enclosed by the graphs of \(f\) and \(g\).
• 16N.2.sl.TZ0.6a: Use the model to find the volume of the barrel.
• 16N.2.sl.TZ0.6b: The empty barrel is being filled with water. The volume \(V{\text{ }}{{\text{m}}^3}\) of water in...
• 16N.2.sl.TZ0.9a: Find the initial velocity of \(P\).
• 16N.2.sl.TZ0.9b: Find the value of \(p\).
• 16N.2.sl.TZ0.9c: (i)     Find the value of \(q\). (ii)     Hence, find the speed of P when \(t = q\).
• 16N.2.sl.TZ0.9d: (i)     Find the total distance travelled by P between \(t = 1\) and \(t = p\). (ii)     Hence...
• 17M.1.sl.TZ1.10a: Show that \(\cos \theta  = \frac{3}{4}\).
• 17M.1.sl.TZ1.10b: Given that \(\tan \theta  > 0\), find \(\tan \theta \).
• 17M.1.sl.TZ1.10c: Let \(y = \frac{1}{{\cos x}}\), for \(0 < x < \frac{\pi }{2}\). The graph of \(y\)between...
• 17M.1.sl.TZ2.10a.i: Write down \(f'(x)\).
• 17M.1.sl.TZ2.10a.ii: Find the gradient of \(L\).
• 17M.1.sl.TZ2.10b: Show that the \(x\)-coordinate of B is \( - \frac{k}{2}\).
• 17M.1.sl.TZ2.10c: Find the area of triangle ABC, giving your answer in terms of \(k\).
• 17M.1.sl.TZ2.10d: Given that the area of triangle ABC is \(p\) times the area of \(R\), find the value of \(p\).
• 17M.2.sl.TZ1.7a.ii: Find the total distance travelled by P, for \(0 \leqslant t \leqslant 8\).
• 17M.2.sl.TZ1.10a.i: Write down the value of \(q\);
• 17M.2.sl.TZ1.10a.ii: Write down the value of \(h\);
• 17M.2.sl.TZ1.10a.iii: Write down the value of \(k\).
• 17M.2.sl.TZ1.10b.i: Find \(\int_{0.111}^{3.31} {\left( {h(x) - x} \right){\text{d}}x} \).
• 17M.2.sl.TZ1.10b.ii: Hence, find the area of the region enclosed by the graphs of \(h\) and \({h^{ - 1}}\).
• 17M.2.sl.TZ1.10c: Let \(d\) be the vertical distance from a point on the graph of \(h\) to the line \(y = x\)....
• 17M.2.sl.TZ2.7: Note:     In this question, distance is in metres and time is in seconds.   A particle moves...
• 17M.2.sl.TZ2.8a: Find the value of \(p\).
• 17M.2.sl.TZ2.8b.i: Write down the coordinates of A.
• 17M.2.sl.TZ2.8b.ii: Write down the rate of change of \(f\) at A.
• 17M.2.sl.TZ2.8c.i: Find the coordinates of B.
• 17M.2.sl.TZ2.8c.ii: Find the the rate of change of \(f\) at B.
• 17M.2.sl.TZ2.8d: Let \(R\) be the region enclosed by the graph of \(f\) , the \(x\)-axis, the line \(x = b\) and...
• 17N.1.sl.TZ0.8a: Show that \(f’(1) = 1\).
• 17N.1.sl.TZ0.8b: Find the equation of \(L\) in the form \(y = ax + b\).
• 17N.1.sl.TZ0.8d: Find the area of the region enclosed by the graph of \(f\) and the line \(L\).
• 17N.1.sl.TZ0.8c: Find the \(x\)-coordinate of Q.
• 17N.2.sl.TZ0.5a: Find the value of \(p\).
• 17N.2.sl.TZ0.5b: The following diagram shows part of the graph of \(f\). The region enclosed by the graph of...
• 17N.2.sl.TZ0.9a: Write down the values of \(t\) when \(a = 0\).
• 17N.2.sl.TZ0.9b: Hence or otherwise, find all possible values of \(t\) for which the velocity of P is decreasing.
• 17N.2.sl.TZ0.9c: Find an expression for the velocity of P at time \(t\).
• 17N.2.sl.TZ0.9d: Find the total distance travelled by P when its velocity is increasing.
• 18M.1.sl.TZ1.5a: Find \(\int {{{\left( {f\left( x \right)} \right)}^2}{\text{d}}x} \).
• 18M.1.sl.TZ1.5b: Part of the graph of f is shown in the following diagram. The shaded region R is enclosed by...
• 18M.2.sl.TZ1.4a: Write down the coordinates of the vertex of the graph of g.
• 18M.2.sl.TZ1.4b: On the grid above, sketch the graph of g for −2 ≤ x ≤ 4.
• 18M.2.sl.TZ1.4c: Find the area of the region enclosed by the graphs of f and g.
• 18M.1.sl.TZ2.2a: Find \(\int {\left( {6{x^2} - 3x} \right){\text{d}}x} \).
• 18M.1.sl.TZ2.2b: Find the area of the region enclosed by the graph of \(f\), the x-axis and the lines x = 1 and x...
• 18M.2.sl.TZ2.3a: Find the x-intercept of the graph of \(f\).
• 18M.2.sl.TZ2.3b: The region enclosed by the graph of \(f\), the y-axis and the x-axis is rotated 360° about the...
• 18M.2.sl.TZ2.9a: Find the initial velocity of P.
• 18M.2.sl.TZ2.9b: Find the maximum speed of P.
• 18M.2.sl.TZ2.9c: Write down the number of times that the acceleration of P is 0 m s−2 .
• 18M.2.sl.TZ2.9d: Find the acceleration of P when it changes direction.
• 18M.2.sl.TZ2.9e: Find the total distance travelled by P.

6.6

• 12N.2.sl.TZ0.7a: On the grid below, sketch the graph of \(s\) .
• 12N.2.sl.TZ0.7b: Find the maximum velocity of the particle.
• 12M.2.sl.TZ2.5a: Find the acceleration of the particle after 2.7 seconds.
• 12M.2.sl.TZ2.5b: Find the displacement of the particle after 1.3 seconds.
• 08N.1.sl.TZ0.9a: Find the acceleration of the particle at \(t = 0\) .
• 08N.1.sl.TZ0.9b: Find the velocity, v, at time t, given that the initial velocity of the particle is...
• 08N.1.sl.TZ0.9c: Find \(\int_0^3 {v{\rm{d}}t} \) , giving your answer in the form \(p - q\cos 3\) .
• 08N.1.sl.TZ0.9d: What information does the answer to part (c) give about the motion of the particle?
• 08M.1.sl.TZ1.6: A particle moves along a straight line so that its velocity, \(v{\text{ m}}{{\text{s}}^{ -...
• 12M.1.sl.TZ1.10a: Find \(s'(t)\) .
• 12M.1.sl.TZ1.10b: In this interval, there are only two values of t for which the object is not moving. One value is...
• 12M.1.sl.TZ1.10c: Show that \(s'(t) > 0\) between these two values of t .
• 12M.1.sl.TZ1.10d: Find the distance travelled between these two values of t .
• 09M.1.sl.TZ1.4a: Complete the following table by noting which graph A, B or C corresponds to each function.
• 09M.1.sl.TZ1.4b: Write down the value of t when the velocity is greatest.
• 09M.1.sl.TZ2.11a: (i)     If \(s = 100\) when \(t = 0\) , find an expression for s in terms of a and t. (ii)    If...
• 09M.1.sl.TZ2.11b: A train M slows down so that it comes to a stop at the station. (i)     Find the time it takes...
• 09M.1.sl.TZ2.11c: For a different train N, the value of a is 4. Show that this train will stop before it reaches...
• 10N.2.sl.TZ0.2a: On the grid below, sketch the graph of v , clearly indicating the maximum point.
• 10N.2.sl.TZ0.2b(i) and (ii): (i)     Write down an expression for d . (ii)    Hence, write down the value of d .
• 10M.2.sl.TZ1.6: The acceleration,  \(a{\text{ m}}{{\text{s}}^{ - 2}}\), of a particle at time t seconds is given...
• SPNone.1.sl.TZ0.3a: Write down the car’s velocity at \(t = 3\) .
• SPNone.1.sl.TZ0.3b: Find the car’s acceleration at \(t = 1.5\) .
• SPNone.1.sl.TZ0.3c: Find the total distance travelled.
• 11N.2.sl.TZ0.7a: Find \(v(t)\) , giving your answer in the form \(a{(t - b)^2} + c\) .
• 11N.2.sl.TZ0.7b: A particle moves along a straight line so that its velocity in ms−1 , at time t seconds, is given...
• 11M.1.sl.TZ1.10a: Write down the velocity of the particle when \(t = 0\) .
• 11M.1.sl.TZ1.10b(i) and (ii): When \(t = k\) , the acceleration is zero. (i)     Show that \(k = \frac{\pi }{4}\) . (ii)   ...
• 11M.1.sl.TZ1.10c: When \(t < \frac{\pi }{4}\) , \(\frac{{{\rm{d}}v}}{{{\rm{d}}t}} > 0\) and when...
• 11M.1.sl.TZ1.10d(i) and (ii): Let d be the distance travelled by the particle for \(0 \le t \le 1\) . (i)     Write down an...
• 11M.1.sl.TZ2.9a: Write down \(f(x)\) in the form \(f(x) = - 10(x - p)(x - q)\) .
• 11M.1.sl.TZ2.9b: Find another expression for \(f(x)\) in the form \(f(x) = - 10{(x - h)^2} + k\) .
• 11M.1.sl.TZ2.9c: Show that \(f(x)\) can also be written in the form \(f(x) = 240 + 20x - 10{x^2}\) .
• 11M.1.sl.TZ2.9d(i) and (ii): A particle moves along a straight line so that its velocity, \(v{\text{ m}}{{\text{s}}^{ - 1}}\)...
• 13M.1.sl.TZ2.6: A rocket moving in a straight line has velocity \(v\) km s–1 and displacement \(s\) km at time...
• 13M.2.sl.TZ1.5b.ii: Write down the positive \(t\)-intercept.
• 14M.2.sl.TZ1.6: Ramiro and Lautaro are travelling from Buenos Aires to El Moro. Ramiro travels in a vehicle...
• 14M.2.sl.TZ2.9a: Find the velocity of the particle when \(t = 1\).
• 14M.2.sl.TZ2.9b: Find the value of \(t\) for which the particle is at rest.
• 14M.2.sl.TZ2.9c: Find the total distance the particle travels during the first three seconds.
• 14M.2.sl.TZ2.9d: Show that the acceleration of the particle is given by \(a = 6t{({t^2} - 4)^2}\).
• 14M.2.sl.TZ2.9e: Find all possible values of \(t\) for which the velocity and acceleration are both positive...
• 13N.2.sl.TZ0.5b: Find the distance travelled by the particle in the first three seconds.
• 13N.2.sl.TZ0.5c: Find the velocity of the particle when its acceleration is zero.
• 14N.2.sl.TZ0.7a: Find the distance travelled by the particle for \(0 \le t \le\ \frac{\pi }{2}\).
• 15N.2.sl.TZ0.6a: Find the value of \(t\) when the particle is at rest.
• 15N.2.sl.TZ0.6b: Find the value of \(t\) when the acceleration of the particle is \(0\).
• 16M.2.sl.TZ1.9a: Find the displacement of P from O after 5 seconds.
• 16M.2.sl.TZ1.9b: Find when P is first at rest.
• 16M.2.sl.TZ1.9c: Write down the number of times P changes direction.
• 16M.2.sl.TZ1.9d: Find the acceleration of P after 3 seconds.
• 16M.2.sl.TZ1.9e: Find the maximum speed of P.
• 16M.2.sl.TZ2.7: A particle moves in a straight line. Its velocity \(v{\text{ m}}\,{{\text{s}}^{ - 1}}\) after...
• 16N.2.sl.TZ0.9a: Find the initial velocity of \(P\).
• 16N.2.sl.TZ0.9b: Find the value of \(p\).
• 16N.2.sl.TZ0.9c: (i)     Find the value of \(q\). (ii)     Hence, find the speed of P when \(t = q\).
• 16N.2.sl.TZ0.9d: (i)     Find the total distance travelled by P between \(t = 1\) and \(t = p\). (ii)     Hence...
• 17M.2.sl.TZ1.7a.i: Write down the first value of \(t\) at which P changes direction.
• 17M.2.sl.TZ1.7a.ii: Find the total distance travelled by P, for \(0 \leqslant t \leqslant 8\).
• 17M.2.sl.TZ1.7b: A second particle Q also moves along a straight line. Its velocity,...
• 17M.2.sl.TZ2.7: Note:     In this question, distance is in metres and time is in seconds.   A particle moves...
• 17N.2.sl.TZ0.9a: Write down the values of \(t\) when \(a = 0\).
• 17N.2.sl.TZ0.9b: Hence or otherwise, find all possible values of \(t\) for which the velocity of P is decreasing.
• 17N.2.sl.TZ0.9c: Find an expression for the velocity of P at time \(t\).
• 17N.2.sl.TZ0.9d: Find the total distance travelled by P when its velocity is increasing.
• 18M.2.sl.TZ1.10a: Find the coordinates of A.
• 18M.2.sl.TZ1.10b.i: For the graph of \(f\), write down the amplitude.
• 18M.2.sl.TZ1.10c: Hence, write \(f\left( x \right)\) in the form \(p\,\,{\text{cos}}\,\left( {x + r} \right)\).
• 18M.2.sl.TZ1.10d: Find the maximum speed of the ball.
• 18M.2.sl.TZ1.10e: Find the first time when the ball’s speed is changing at a rate of 2 cm s−2.
• 18M.2.sl.TZ1.10b.ii: For the graph of \(f\), write down the period.
• 18M.2.sl.TZ2.9a: Find the initial velocity of P.
• 18M.2.sl.TZ2.9b: Find the maximum speed of P.
• 18M.2.sl.TZ2.9c: Write down the number of times that the acceleration of P is 0 m s−2 .
• 18M.2.sl.TZ2.9d: Find the acceleration of P when it changes direction.
• 18M.2.sl.TZ2.9e: Find the total distance travelled by P.