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...