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...