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...