DP Mathematics SL Questionbank

• Syllabus

6.1

Sub sections and their related questions

Informal ideas of limit and convergence.

• 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.TZ0.10a: (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.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...

Limit notation.

• 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.TZ0.10a: (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.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...

Definition of derivative from first principles as \(f'\left( x \right) = {\mathop {\lim }\limits_{h \to 0 } } \left( {\frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}} \right)\) .

• 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...
• 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.TZ0.10a: (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.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...

Derivative interpreted as gradient function and as rate of change.

• 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 \(a\) .
• 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...
• 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.TZ1.10c: 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.
• 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.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...
• 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.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.
• 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.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.
• 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.TZ0.10a: (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.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...

Tangents and normals, and their equations.

• 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.
• 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.TZ2.6b: The normal to the curve at P cuts the x-axis at R. Find the coordinates 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...
• 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.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.
• 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...
• 15M.1.sl.TZ1.9d: Find the equation of the tangent to the curve of \(f\) at \(( - 2,{\text{ }}1)\), giving your...
• 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...
• 16N.2.sl.TZ0.2b: (i)     sketch the graph of \(f\), clearly indicating the point A; (ii)    sketch the tangent to...
• 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 \(p\).
• 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...