DP Mathematics HL Questionbank

• Syllabus

6.2

Sub sections and their related questions

Derivatives of ${x^n}$ , $\sin x$ , $\cos x$ , $\tan x$ , ${{\text{e}}^x}$ and \$\ln x$ .

• 12M.1.hl.TZ1.12a: Show that $f'(x) = \frac{1}{2}{x^{ - \frac{1}{2}}}{(1 - x)^{ - \frac{3}{2}}}$ and deduce that f...
• 12N.1.hl.TZ0.8a: Find the gradient of the tangent to the curve at the point $(\pi ,{\text{ }}\pi )$ .
• 12N.2.hl.TZ0.12c: Let a = 3k and b = k . Find $\frac{{{\text{d}}L}}{{{\text{d}}\alpha }}$.
• SPNone.1.hl.TZ0.9a: (i)     Find an expression for $f'(x)$ . (ii)     Given that the equation $f'(x) = 0$ has...
• SPNone.1.hl.TZ0.12a: Show that $f''(x) = 2{{\text{e}}^x}\sin \left( {x + \frac{\pi }{2}} \right)$ .
• SPNone.1.hl.TZ0.12b: Obtain a similar expression for ${f^{(4)}}(x)$ .
• SPNone.1.hl.TZ0.13b: Show that f is a one-to-one function.
• SPNone.2.hl.TZ0.13a: Obtain an expression for $f'(x)$ .
• SPNone.3ca.hl.TZ0.1a: Show that $f''(x) = - \frac{1}{{(1 + \sin x)}}$ .
• 13M.2.hl.TZ1.13a: Verify that this is true for $f(x) = {x^3} + 1$ at x = 2.
• 13M.2.hl.TZ1.13b: Given that $g(x) = x{{\text{e}}^{{x^2}}}$, show that $g'(x) > 0$ for all values of x.
• 13M.2.hl.TZ1.13c: Using the result given at the start of the question, find the value of the gradient function of...
• 13M.2.hl.TZ2.13d: Show that...
• 11N.1.hl.TZ0.8b: Find the value of $\theta$ for which $\frac{{{\text{d}}t}}{{{\text{d}}\theta }} = 0$.
• 11N.3ca.hl.TZ0.5a: Given that $y = \ln \left( {\frac{{1 + {{\text{e}}^{ - x}}}}{2}} \right)$, show that...
• 11M.1.hl.TZ1.12a: (i)     Solve the equation $f'(x) = 0$ . (ii)     Hence show the graph of $f$ has a local...
• 11M.1.hl.TZ1.12b: Show that there is a point of inflexion on the graph and determine its coordinates.
• 11M.1.hl.TZ1.12c: Sketch the graph of $y = f(x)$ , indicating clearly the asymptote, x-intercept and the local...
• 14M.1.hl.TZ2.14c: Given that $f(x) = h(x) + h \circ g(x)$, (i)     find $f'(x)$ in simplified form; (ii)    ...
• 13N.1.hl.TZ0.10a(i)(ii): (i)     Find an expression for $f'(x)$. (ii)     Hence determine the coordinates of the point...
• 14N.2.hl.TZ0.10b: (i)     State $\frac{{{\text{d}}A}}{{{\text{d}}x}}$. (ii)     Verify that...
• 15M.1.hl.TZ1.11a: Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$.
• 15M.1.hl.TZ2.11c: Let $y = g \circ f(x)$, find an exact value for $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ at the...
• 15M.2.hl.TZ1.6: A function $f$ is defined by $f(x) = {x^3} + {{\text{e}}^x} + 1,{\text{ }}x \in \mathbb{R}$....
• 15N.3ca.hl.TZ0.2a: Show that $f''(x) = 2\left( {f'(x) - f(x)} \right)$.
• 15N.1.hl.TZ0.4a: Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$.
• 15N.1.hl.TZ0.8b: Consider $f(x) = \sin (ax)$ where $a$ is a constant. Prove by mathematical induction that...

Differentiation of sums and multiples of functions.

• 12M.1.hl.TZ1.12a: Show that $f'(x) = \frac{1}{2}{x^{ - \frac{1}{2}}}{(1 - x)^{ - \frac{3}{2}}}$ and deduce that f...
• 12N.1.hl.TZ0.8a: Find the gradient of the tangent to the curve at the point $(\pi ,{\text{ }}\pi )$ .
• 12N.2.hl.TZ0.12c: Let a = 3k and b = k . Find $\frac{{{\text{d}}L}}{{{\text{d}}\alpha }}$.
• 11M.1.hl.TZ1.12a: (i)     Solve the equation $f'(x) = 0$ . (ii)     Hence show the graph of $f$ has a local...
• 11M.1.hl.TZ1.12b: Show that there is a point of inflexion on the graph and determine its coordinates.
• 11M.1.hl.TZ1.12c: Sketch the graph of $y = f(x)$ , indicating clearly the asymptote, x-intercept and the local...
• 13N.1.hl.TZ0.10a(i)(ii): (i)     Find an expression for $f'(x)$. (ii)     Hence determine the coordinates of the point...
• 14N.2.hl.TZ0.10b: (i)     State $\frac{{{\text{d}}A}}{{{\text{d}}x}}$. (ii)     Verify that...

The product and quotient rules.

• 12M.1.hl.TZ1.12a: Show that $f'(x) = \frac{1}{2}{x^{ - \frac{1}{2}}}{(1 - x)^{ - \frac{3}{2}}}$ and deduce that f...
• 12N.1.hl.TZ0.8a: Find the gradient of the tangent to the curve at the point $(\pi ,{\text{ }}\pi )$ .
• 12N.2.hl.TZ0.12c: Let a = 3k and b = k . Find $\frac{{{\text{d}}L}}{{{\text{d}}\alpha }}$.
• 08N.2.hl.TZ0.12: The function f is defined by...
• 13M.1.hl.TZ2.5a: Show that...
• 13M.1.hl.TZ2.12b: Hence show that $f'(x) > 0$ on D.
• 11M.1.hl.TZ1.12a: (i)     Solve the equation $f'(x) = 0$ . (ii)     Hence show the graph of $f$ has a local...
• 11M.1.hl.TZ1.12b: Show that there is a point of inflexion on the graph and determine its coordinates.
• 11M.1.hl.TZ1.12c: Sketch the graph of $y = f(x)$ , indicating clearly the asymptote, x-intercept and the local...
• 14M.1.hl.TZ1.11a: Show that $f'(x) = \frac{{1 - \ln x}}{{{x^2}}}$.
• 14M.2.hl.TZ1.10d: Find the equation of the line ${L_2}$.
• 14M.1.hl.TZ2.13c: Find $f''(x)$ expressing your answer in the form $\frac{{p(x)}}{{{{({x^2} + 1)}^3}}}$, where...
• 13N.1.hl.TZ0.10a(i)(ii): (i)     Find an expression for $f'(x)$. (ii)     Hence determine the coordinates of the point...
• 14M.1.hl.TZ2.13a: Find $f'(x)$.
• 14M.2.hl.TZ1.10b: (i)     Find $f'(x)$. (ii)     Show that the curve has exactly one point where its tangent is...
• 14M.2.hl.TZ1.10c: Find the equation of ${L_1}$, the normal to the curve at the point where it crosses the y-axis.
• 14N.1.hl.TZ0.5: A tranquilizer is injected into a muscle from which it enters the bloodstream. The concentration...
• 14N.1.hl.TZ0.7a: $p'(3)$;
• 15M.1.hl.TZ1.11a: Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$.
• 15M.1.hl.TZ2.11c: Let $y = g \circ f(x)$, find an exact value for $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ at the...
• 15N.3ca.hl.TZ0.2a: Show that $f''(x) = 2\left( {f'(x) - f(x)} \right)$.
• 15N.1.hl.TZ0.12b: Find $f'(x)$.
• 15N.2.hl.TZ0.13a: Find $f''(x)$.
• 16M.2.hl.TZ1.12e: Given that $v = {y^3},{\text{ }}y > 0$, find $\frac{{{\text{d}}v}}{{{\text{d}}x}}$ at...
• 16M.1.hl.TZ2.11b: (i)     Given that $\frac{{{\text{d}}V}}{{{\text{d}}h}} = \pi {(3\cos 2h + 4)^2}$, find an...
• 16M.1.hl.TZ1.9: A curve is given by the equation $y = \sin (\pi \cos x)$. Find the coordinates of all the...
• 16M.2.hl.TZ2.12c: (i)     Show that $t'(x) = \frac{{{{[f(x)]}^2} - {{[g(x)]}^2}}}{{{{[f(x)]}^2}}}$ for...
• 16N.1.hl.TZ0.11a: Find an expression for $\frac{{{\text{d}}y}}{{{\text{d}}x}}$.
• 16N.2.hl.TZ0.6: An earth satellite moves in a path that can be described by the curve...
• 16N.2.hl.TZ0.10c: Show that $f'(x) =  - \frac{{3{{\text{e}}^x}}}{{{{(2{{\text{e}}^x} - 1)}^2}}}$.
• 17M.2.hl.TZ1.8b: Calculate $\frac{{{\text{d}}\theta }}{{{\text{d}}t}}$ when $\theta = \frac{\pi }{3}$.
• 17M.2.hl.TZ1.12f: Find $g'(x)$.
• 17M.2.hl.TZ1.12g.i: Hence, show that there are no solutions to $g'(x) = 0$;
• 17M.2.hl.TZ1.12g.ii: Hence, show that there are no solutions to $({g^{ - 1}})'(x) = 0$.
• 18M.1.hl.TZ1.2a: Find $\frac{{{\text{d}}y}}{{{\text{d}}\theta }}$
• 18M.1.hl.TZ1.2b: Hence find the values of θ for which $\frac{{{\text{d}}y}}{{{\text{d}}\theta }} = 2y$.
• 18M.1.hl.TZ1.7a: Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$.
• 18M.1.hl.TZ2.6a.i: Find $f'\left( x \right)$.
• 18M.1.hl.TZ2.6a.ii: Find $g'\left( x \right)$.

The chain rule for composite functions.

• 12N.1.hl.TZ0.8a: Find the gradient of the tangent to the curve at the point $(\pi ,{\text{ }}\pi )$ .
• 08N.2.hl.TZ0.8: If $y = \ln \left( {\frac{1}{3}(1 + {{\text{e}}^{ - 2x}})} \right)$, show that...
• 09M.2.hl.TZ2.3: (a)     Differentiate $f(x) = \arcsin x + 2\sqrt {1 - {x^2}}$ , $x \in [ - 1, 1]$ . (b)    ...
• 14M.1.hl.TZ1.9: A curve has equation $\arctan {x^2} + \arctan {y^2} = \frac{\pi }{4}$. (a)     Find...
• 14M.2.hl.TZ2.12: Engineers need to lay pipes to connect two cities A and B that are separated by a river of width...
• 14N.1.hl.TZ0.7b: $h'(2)$.
• 15M.1.hl.TZ1.11a: Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$.
• 15N.1.hl.TZ0.4a: Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$.
• 15N.1.hl.TZ0.12b: Find $f'(x)$.
• 15N.2.hl.TZ0.13a: Find $f''(x)$.
• 16M.2.hl.TZ1.12e: Given that $v = {y^3},{\text{ }}y > 0$, find $\frac{{{\text{d}}v}}{{{\text{d}}x}}$ at...
• 16M.1.hl.TZ2.11b: (i)     Given that $\frac{{{\text{d}}V}}{{{\text{d}}h}} = \pi {(3\cos 2h + 4)^2}$, find an...
• 16M.1.hl.TZ1.9: A curve is given by the equation $y = \sin (\pi \cos x)$. Find the coordinates of all the...
• 16M.2.hl.TZ2.12c: (i)     Show that $t'(x) = \frac{{{{[f(x)]}^2} - {{[g(x)]}^2}}}{{{{[f(x)]}^2}}}$ for...
• 16N.2.hl.TZ0.6: An earth satellite moves in a path that can be described by the curve...
• 16N.2.hl.TZ0.10c: Show that $f'(x) =  - \frac{{3{{\text{e}}^x}}}{{{{(2{{\text{e}}^x} - 1)}^2}}}$.
• 17M.2.hl.TZ1.8b: Calculate $\frac{{{\text{d}}\theta }}{{{\text{d}}t}}$ when $\theta = \frac{\pi }{3}$.
• 17M.2.hl.TZ1.12f: Find $g'(x)$.
• 17M.2.hl.TZ1.12g.i: Hence, show that there are no solutions to $g'(x) = 0$;
• 17M.2.hl.TZ1.12g.ii: Hence, show that there are no solutions to $({g^{ - 1}})'(x) = 0$.
• 18M.1.hl.TZ1.2a: Find $\frac{{{\text{d}}y}}{{{\text{d}}\theta }}$
• 18M.1.hl.TZ1.2b: Hence find the values of θ for which $\frac{{{\text{d}}y}}{{{\text{d}}\theta }} = 2y$.
• 18M.1.hl.TZ1.7a: Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$.
• 18M.1.hl.TZ2.6a.i: Find $f'\left( x \right)$.
• 18M.1.hl.TZ2.6a.ii: Find $g'\left( x \right)$.

Related rates of change.

• 12N.1.hl.TZ0.8a: Find the gradient of the tangent to the curve at the point $(\pi ,{\text{ }}\pi )$ .
• 11M.2.hl.TZ2.9: A rocket is rising vertically at a speed of $300{\text{ m}}{{\text{s}}^{ - 1}}$ when it is 800...
• SPNone.2.hl.TZ0.9: A ladder of length 10 m on horizontal ground rests against a vertical wall. The bottom of the...
• 13M.1.hl.TZ1.5: Paint is poured into a tray where it forms a circular pool with a uniform thickness of 0.5 cm. If...
• 10M.2.hl.TZ2.10: A lighthouse L is located offshore, 500 metres from the nearest point P on a long straight...
• 13M.2.hl.TZ2.13f: The point P moves across the street with speed $0.5{\text{ m}}{{\text{s}}^{ - 1}}$. Determine...
• 11N.2.hl.TZ0.9: A stalactite has the shape of a circular cone. Its height is 200 mm and is increasing at a rate...
• 11M.2.hl.TZ1.14b: If the water in the glass evaporates at the rate of 3 cm3 per hour for each cm2 of exposed...
• 09N.2.hl.TZ0.12: (a)     The circular Ferris wheel has a radius of 10 metres and is revolving at a rate of 3...
• 09M.2.hl.TZ1.9: (a)     Given that $\frac{{{\text{d}}y}}{{{\text{d}}t}} = 0.001r$ , show that...
• 09M.2.hl.TZ2.10: (a)     show that the rate of change of ${\rm{H}}\hat {\text{P}}{\text{Q}}$ is $0.16$ radians...
• 14M.1.hl.TZ1.9: A curve has equation $\arctan {x^2} + \arctan {y^2} = \frac{\pi }{4}$. (a)     Find...
• 14M.2.hl.TZ2.9: Sand is being poured to form a cone of height $h$ cm and base radius $r$ cm. The height...
• 14N.2.hl.TZ0.4: Two cyclists are at the same road intersection. One cyclist travels north at...
• 15M.2.hl.TZ1.5: A bicycle inner tube can be considered as a joined up cylinder of fixed length $200$ cm and...
• 17M.2.hl.TZ1.8b: Calculate $\frac{{{\text{d}}\theta }}{{{\text{d}}t}}$ when $\theta = \frac{\pi }{3}$.

Implicit differentiation.

• 12M.1.hl.TZ1.9: The curve C has equation $2{x^2} + {y^2} = 18$. Determine the coordinates of the four points on...
• 12M.1.hl.TZ2.8: Let ${x^3}y = a\sin nx$ . Using implicit differentiation, show...
• 12N.1.hl.TZ0.8a: Find the gradient of the tangent to the curve at the point $(\pi ,{\text{ }}\pi )$ .
• 12N.2.hl.TZ0.6: A particle moves along a straight line so that after t seconds its displacement s , in...
• 08M.2.hl.TZ1.6: Find the gradient of the tangent to the curve ${x^3}{y^2} = \cos (\pi y)$ at the point (−1, 1) .
• 08M.1.hl.TZ2.5: Consider the curve with equation ${x^2} + xy + {y^2} = 3$. (a)     Find in terms of k, the...
• 11M.2.hl.TZ2.10: The point P, with coordinates $(p,{\text{ }}q)$ , lies on the graph of...
• 09M.1.hl.TZ2.5: Consider the part of the curve $4{x^2} + {y^2} = 4$ shown in the diagram below.     (a)  ...
• 13M.1.hl.TZ1.7: A curve is defined by the equation $8y\ln x - 2{x^2} + 4{y^2} = 7$. Find the equation of the...
• 13M.1.hl.TZ2.8a: Express $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ in terms of x and y.
• 13M.1.hl.TZ2.8b: Find the value of $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ at the point on C where y = 1 and...
• 11M.1.hl.TZ1.9: Show that the points (0, 0) and ($\sqrt {2\pi }$ , $- \sqrt {2\pi }$) on the curve...
• 09N.2.hl.TZ0.8: Find the gradient of the curve...
• 14M.1.hl.TZ1.9: A curve has equation $\arctan {x^2} + \arctan {y^2} = \frac{\pi }{4}$. (a)     Find...
• 14M.2.hl.TZ2.10a: Use implicit differentiation to find an expression for $\frac{{{\text{d}}y}}{{{\text{d}}x}}$.
• 13N.1.hl.TZ0.5: A curve has equation ${x^3}{y^2} + {x^3} - {y^3} + 9y = 0$. Find the coordinates of the three...
• 15M.2.hl.TZ2.11a: Show that $\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{5y - 2x}}{{2y - 5x}}$.
• 15N.1.hl.TZ0.7a: Show that there is no point where the tangent to the curve is horizontal.
• 16M.2.hl.TZ1.12b: Show that $\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{2y - {{\text{e}}^x}}}{{2(y - x)}}$.
• 16M.2.hl.TZ2.7a: Use implicit differentiation to show that...
• 16N.1.hl.TZ0.9a: Find an expression for $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ in terms of $x$ and $y$.
• 17M.2.hl.TZ1.2a: Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ in terms of $x$ and $y$.
• 17M.2.hl.TZ2.2a: Find the equation of the normal to the curve at the point $\left( {1,{\text{ }}\sqrt 3 } \right)$.
• 17N.1.hl.TZ0.7: The folium of Descartes is a curve defined by the equation ${x^3} + {y^3} - 3xy = 0$, shown in...
• 18M.2.hl.TZ2.11a: Show...

Derivatives of $\sec x$ , $\csc x$ , $\cot x$ , ${a^x}$ , ${\log _a}x$ , $\arcsin x$ , $\arccos x$ and $\arctan x$ .

• 14M.1.hl.TZ1.9: A curve has equation $\arctan {x^2} + \arctan {y^2} = \frac{\pi }{4}$. (a)     Find...