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.