DP Mathematics SL Questionbank

• Syllabus

6.3

Sub sections and their related questions

Local maximum and minimum points.

• 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 ?
• 08M.1.sl.TZ1.8b: Find the x-coordinate of M.
• 08M.2.sl.TZ1.5b: Complete the table, for the graph of $y = f(x)$ .
• 08M.1.sl.TZ2.10d: Find the value of $\theta$ when S is a local minimum, justifying that it is a minimum.
• 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.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 $p$ .
• 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.
• 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 $p$.
• 14N.1.sl.TZ0.9a: Find the $x$-coordinate of $A$.
• 15N.1.sl.TZ0.10a: Explain why the graph of $f$ has a local minimum when $x = 5$.
• 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.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.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...
• 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.

Testing for maximum or minimum.

• 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...
• 08M.1.sl.TZ2.10d: Find the value of $\theta$ when S is a local minimum, justifying that it is a minimum.
• 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.TZ0.9b: The second derivative $f''(x) = \frac{{40(3{x^2} + 4)}}{{{{({x^2} - 4)}^3}}}$ . Use this...
• SPNone.1.sl.TZ0.10b(i) and (ii): Hence (i)     show that $q = - 2$ ; (ii)    verify that A is a minimum point.
• 15M.1.sl.TZ1.9e: Given that $f'( - 1) = 0$, explain why the graph of $f$ has a local maximum when $x =  - 1$.
• 15N.1.sl.TZ0.10a: Explain why the graph of $f$ has a local minimum when $x = 5$.
• 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.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.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...
• 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.

Points of inflexion with zero and non-zero gradients.

• 08N.1.sl.TZ0.6c: Show that B corresponds to a point of inflexion on the graph of f .
• 08M.1.sl.TZ1.8c: Find the x-coordinate of N.
• 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).
• 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...
• 09N.1.sl.TZ0.9b: The second derivative $f''(x) = \frac{{40(3{x^2} + 4)}}{{{{({x^2} - 4)}^3}}}$ . Use this...
• 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.
• 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.
• 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.TZ2.10b: Explain why P is a point of inflexion.
• 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.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.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.TZ2.8c.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...
• 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.

Graphical behaviour of functions, including the relationship between the graphs of $f$ , ${f'}$ and ${f''}$ .

• 08M.2.sl.TZ1.5a: On the grid below, sketch a graph of $y = f''(x)$ , clearly indicating the x-intercept.
• 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.
• 09N.1.sl.TZ0.9c: Describe the behaviour of the graph of $f$ for large $|x|$ .
• 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.
• 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.
• 13M.1.sl.TZ1.10d: There is a point of inflexion on the graph of $f$ at $x = \sqrt[4]{3}$...
• 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:...
• 15M.1.sl.TZ1.9b: The graph of $f$ has a point of inflexion when $x = 1$. Show that $k = 3$.
• 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.
• 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.
• 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.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.

Optimization.

• 08M.1.sl.TZ2.10e: Find a value of $\theta$ for which S has its greatest value.
• 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.TZ1.8c: (i)     Find $\frac{{{\rm{d}}A}}{{{\rm{d}}\theta }}$ . (ii)    Hence, find the exact value of...
• 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.2.sl.TZ1.9e: Find the maximum rate of change of $f$ .
• 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...
• 14N.1.sl.TZ0.10d: There is a minimum value for $d$. Find the value of $a$ that gives this minimum value.
• 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...
• 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.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 $y = x$....
• 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.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 $\pi$.

Applications.

• 13N.2.sl.TZ0.7a: Let ${\text{OP}} = x$. (i)     Find ${\text{PQ}}$, giving your answer in terms of...