DP Mathematics HL Questionbank

• Syllabus

Topic 6 - Core: Calculus

Sub sections and their related questions

6.1

• 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.TZ1.12a: Show that $f'(x) = \frac{1}{2}{x^{ - \frac{1}{2}}}{(1 - x)^{ - \frac{3}{2}}}$ and deduce that f...
• 12M.2.hl.TZ1.1: Given that the graph of $y = {x^3} - 6{x^2} + kx - 4$ has exactly one point at which...
• 12M.2.hl.TZ1.11d: Find the coordinates of the point of intersection of the normals to the graph at the points P and Q.
• 12M.1.hl.TZ2.13a: Using the definition of a derivative as...
• 12M.2.hl.TZ2.6b: Write down the gradient of the curve at x = 1 .
• 12M.2.hl.TZ2.6c: Find the equation of the normal to the curve at x = 1 .
• 08M.2.hl.TZ1.13: A family of cubic functions is defined as...
• 08M.1.hl.TZ2.8: A normal to the graph of $y = \arctan (x - 1)$ , for $x > 0$, has equation...
• 08N.1.hl.TZ0.6: Find the equation of the normal to the curve $5x{y^2} - 2{x^2} = 18$ at the point (1, 2) .
• 11M.1.hl.TZ2.11b: The tangent to C at the point P(1, 2) cuts the x-axis at the point T. Determine the coordinates...
• 11M.1.hl.TZ2.11c: The normal to C at the point P cuts the y-axis at the point N. Find the area of triangle PTN.
• 09M.1.hl.TZ1.7: Consider the functions f and g defined by $f(x) = {2^{\frac{1}{x}}}$ and...
• 09N.1.hl.TZ0.12: A tangent to the graph of $y = \ln x$ passes through the origin. (a)     Sketch the graphs of...
• SPNone.1.hl.TZ0.5b: Show that $f''(0) = 0$ .
• SPNone.1.hl.TZ0.13a: Given that f and its derivative, $f'$ , are continuous for all values in the domain of f , find...
• SPNone.2.hl.TZ0.13d: Find the equation of the normal to the graph of f where x = 0.75 , giving your answer in the form...
• 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...
• 10M.1.hl.TZ2.8: The normal to the curve $x{{\text{e}}^{ - y}} + {{\text{e}}^y} = 1 + x$, at the point (c,...
• 10N.1.hl.TZ0.12b: Consider the...
• 10N.1.hl.TZ0.13: Consider the curve $y = x{{\text{e}}^x}$ and the line \(y = kx,{\text{ }}k \in...
• 10N.2.hl.TZ0.4: Find the equation of the normal to the curve ${x^3}{y^3} - xy = 0$ at the point (1, 1).
• 10N.2.hl.TZ0.10: The line $y = m(x - m)$ is a tangent to the curve $(1 - x)y = 1$. Determine m and the...
• 13M.1.hl.TZ2.5b: Find the equation of the tangent to C at the point $\left( {\frac{\pi }{2},0} \right)$.
• 11N.1.hl.TZ0.6: Given that $y = \frac{1}{{1 - x}}$, use mathematical induction to prove that...
• 11N.1.hl.TZ0.13a: Find the equation of the tangent to C at the point (2, e).
• 09M.2.hl.TZ1.12: (a)     If A, B and C have x-coordinates $a\frac{\pi }{2}$, $b\frac{\pi }{6}$ and...
• 14M.3ca.hl.TZ0.2a: Consider the functions $f(x) = {(\ln x)^2},{\text{ }}x > 1$ and...
• 14M.1.hl.TZ1.11d: The graph of $y = {\text{ }}f(x)$ crosses the $x$-axis at the point A. Find the equation of...
• 14M.2.hl.TZ1.10d: Find the equation of the line ${L_2}$.
• 14M.1.hl.TZ2.8a: Determine whether or not $f$is continuous.
• 14M.2.hl.TZ2.10b: Find the equation of the normal to the curve at the point (1, 1).
• 13N.2.hl.TZ0.13a: (i)   Explain why the inverse function ${f^{ - 1}}$ does not exist. (ii)     Show that the...
• 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.TZ0.11c: The graph of $y = g(x)$ intersects the $x$-axis at the point $Q$. Show that the equation...
• 14N.1.hl.TZ0.11e: A region $R$ is bounded by the graphs of $y = g(x)$, the tangent $T$ and the line...
• 15M.1.hl.TZ2.4a: Determine the values of $x$ for which $f(x)$ is a decreasing function.
• 15M.2.hl.TZ1.9: Find the equation of the normal to the curve...
• 15M.2.hl.TZ2.11b: Find the equation of the normal to the curve at the point $(6,{\text{ }}1)$.
• 15M.2.hl.TZ2.12c: For $t > 5$, the displacement of the particle is given by...
• 15N.3ca.hl.TZ0.2a: Show that $f''(x) = 2\left( {f'(x) - f(x)} \right)$.
• 15N.3ca.hl.TZ0.5a: Show that the tangent to the curve $y = f(x)$ at the point $(1,{\text{ }}0)$ is normal to the...
• 15N.1.hl.TZ0.4b: Determine the equation of the normal to the curve at the point $x = 3$ in the form...
• 15N.1.hl.TZ0.7a: Show that there is no point where the tangent to the curve is horizontal.
• 15N.1.hl.TZ0.7b: Find the coordinates of the points where the tangent to the curve is vertical.
• 15N.1.hl.TZ0.8b: Consider $f(x) = \sin (ax)$ where $a$ is a constant. Prove by mathematical induction that...
• 15N.2.hl.TZ0.13a: Find $f''(x)$.
• 16M.2.hl.TZ1.12a: Find the value of $a$.
• 16M.2.hl.TZ1.12c: Find the equation of the normal to $C$ at the point A.
• 16M.2.hl.TZ1.12d: Find the coordinates of the second point at which the normal found in part (c) intersects $C$.
• 16M.1.hl.TZ2.4: The function $f$ is defined as $f(x) = a{x^2} + bx + c$ where...
• 16M.1.hl.TZ2.11c: (i)     Find $\frac{{{{\text{d}}^2}h}}{{{\text{d}}{t^2}}}$. (ii)     Find the values of...
• 16M.1.hl.TZ1.10: Find the $x$-coordinates of all the points on the curve...
• 16M.2.hl.TZ2.7b: Find the value of $k$.
• 16N.1.hl.TZ0.9b: Find the equations of the tangents to this curve at the points where the curve intersects the...
• 16N.1.hl.TZ0.11b: Show that $\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}} = 2{{\text{e}}^x}\cos x$.
• 16N.1.hl.TZ0.11g: Find the value of the curvature of the graph of $f$ at the local maximum point.
• 16N.1.hl.TZ0.11h: Find the value $\kappa$ for $x = \frac{\pi }{2}$ and comment on its meaning with respect to...
• 17M.1.hl.TZ2.9c: By finding $g'(x)$ explain why $g$ is an increasing function.
• 17M.2.hl.TZ1.2b: Determine the equation of the tangent to $C$ at the point...
• 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.TZ0.11c: Hence or otherwise, find an expression for the derivative of ${f_n}(x)$ with respect to $x$.
• 17N.1.hl.TZ0.11d: Show that, for $n > 1$, the equation of the tangent to the curve $y = {f_n}(x)$ at...
• 17N.2.hl.TZ0.10a.i: Show that the $x$-coordinate of the minimum point on the curve $y = f(x)$ satisfies the...
• 17N.2.hl.TZ0.10a.ii: Determine the values of $x$ for which $f(x)$ is a decreasing function.
• 17N.2.hl.TZ0.10c: Find the coordinates of the point on the graph of $f$ where the normal to the graph is parallel...
• 17N.2.hl.TZ0.11a.i: Determine an expression for $f’(x)$ in terms of $x$.
• 17N.2.hl.TZ0.11a.ii: Sketch a graph of $y = f’(x)$ for $0 \leqslant x < \frac{\pi }{2}$.
• 17N.2.hl.TZ0.11a.iii: Find the $x$-coordinate(s) of the point(s) of inflexion of the graph of $y = f(x)$, labelling...
• 18M.2.hl.TZ1.9b: Find the equation of the normal to the curve at the point P.
• 18M.2.hl.TZ1.9c: The normal at P cuts the curve again at the point Q. Find the $x$-coordinate of Q.
• 18M.2.hl.TZ2.11b.i: Find the coordinates of P and Q.
• 18M.2.hl.TZ2.11b.ii: Given that the gradients of the tangents to C at P and Q are m1 and m2 respectively, show that m1...
• 18M.2.hl.TZ2.11c: Find the coordinates of the three points on C, nearest the origin, where the tangent is parallel...

6.2

• 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.TZ1.12a: Show that $f'(x) = \frac{1}{2}{x^{ - \frac{1}{2}}}{(1 - x)^{ - \frac{3}{2}}}$ and deduce that f...
• 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...
• 12N.2.hl.TZ0.12c: Let a = 3k and b = k . Find $\frac{{{\text{d}}L}}{{{\text{d}}\alpha }}$.
• 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...
• 08N.2.hl.TZ0.8: If $y = \ln \left( {\frac{1}{3}(1 + {{\text{e}}^{ - 2x}})} \right)$, show that...
• 08N.2.hl.TZ0.12: The function f is defined by...
• 11M.2.hl.TZ2.9: A rocket is rising vertically at a speed of $300{\text{ m}}{{\text{s}}^{ - 1}}$ when it is 800...
• 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)  ...
• 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.TZ0.9: A ladder of length 10 m on horizontal ground rests against a vertical wall. The bottom of the...
• 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.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...
• 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.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...
• 10M.2.hl.TZ2.10: A lighthouse L is located offshore, 500 metres from the nearest point P on a long straight...
• 13M.1.hl.TZ2.5a: Show that...
• 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...
• 13M.1.hl.TZ2.12b: Hence show that $f'(x) > 0$ on D.
• 13M.2.hl.TZ2.13d: Show that...
• 13M.2.hl.TZ2.13f: The point P moves across the street with speed $0.5{\text{ m}}{{\text{s}}^{ - 1}}$. Determine...
• 11N.1.hl.TZ0.8b: Find the value of $\theta$ for which $\frac{{{\text{d}}t}}{{{\text{d}}\theta }} = 0$.
• 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...
• 11N.3ca.hl.TZ0.5a: Given that $y = \ln \left( {\frac{{1 + {{\text{e}}^{ - x}}}}{2}} \right)$, show that...
• 11M.1.hl.TZ1.9: Show that the points (0, 0) and ($\sqrt {2\pi }$ , $- \sqrt {2\pi }$) on the curve...
• 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...
• 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.TZ0.8: Find the gradient of the curve...
• 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.TZ2.3: (a)     Differentiate $f(x) = \arcsin x + 2\sqrt {1 - {x^2}}$ , $x \in [ - 1, 1]$ . (b)    ...
• 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.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...
• 14M.1.hl.TZ2.14c: Given that $f(x) = h(x) + h \circ g(x)$, (i)     find $f'(x)$ in simplified form; (ii)    ...
• 14M.2.hl.TZ2.9: Sand is being poured to form a cone of height $h$ cm and base radius $r$ cm. The height...
• 14M.2.hl.TZ2.10a: Use implicit differentiation to find an expression for $\frac{{{\text{d}}y}}{{{\text{d}}x}}$.
• 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...
• 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...
• 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: $p'(3)$;
• 14N.1.hl.TZ0.7b: $h'(2)$.
• 14N.2.hl.TZ0.4: Two cyclists are at the same road intersection. One cyclist travels north at...
• 14N.2.hl.TZ0.10b: (i)     State $\frac{{{\text{d}}A}}{{{\text{d}}x}}$. (ii)     Verify that...
• 15M.1.hl.TZ1.3b: Find $\int {{{\sin }^2}x{\text{d}}x}$.
• 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.TZ1.5: A bicycle inner tube can be considered as a joined up cylinder of fixed length $200$ cm and...
• 15M.2.hl.TZ1.6: A function $f$ is defined by $f(x) = {x^3} + {{\text{e}}^x} + 1,{\text{ }}x \in \mathbb{R}$....
• 15M.2.hl.TZ2.11a: Show that $\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{5y - 2x}}{{2y - 5x}}$.
• 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.TZ0.7a: Show that there is no point where the tangent to the curve is horizontal.
• 15N.1.hl.TZ0.8b: Consider $f(x) = \sin (ax)$ where $a$ is a constant. Prove by mathematical induction that...
• 15N.1.hl.TZ0.12b: Find $f'(x)$.
• 15N.2.hl.TZ0.13a: Find $f''(x)$.
• 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.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.TZ2.7a: Use implicit differentiation to show that...
• 16M.2.hl.TZ2.12c: (i)     Show that $t'(x) = \frac{{{{[f(x)]}^2} - {{[g(x)]}^2}}}{{{{[f(x)]}^2}}}$ for...
• 16N.1.hl.TZ0.9a: Find an expression for $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ in terms of $x$ and $y$.
• 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.TZ1.2a: Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ in terms of $x$ and $y$.
• 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 $({g^{ - 1}})'(x) = 0$.
• 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.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 $g'\left( x \right)$.
• 18M.2.hl.TZ2.11a: Show...

6.3

• 12M.1.hl.TZ1.2b: Hence state the value of (i)     $f'( - 3)$; (ii)     $f'(2.7)$; (iii)  ...
• 12M.1.hl.TZ1.12b: Show that the curve $y = f(x)$ has one point of inflexion, and find its coordinates.
• 12M.2.hl.TZ1.10: A triangle is formed by the three lines $y = 10 - 2x,{\text{ }}y = mx$ and...
• 12M.1.hl.TZ2.7b: On the axes below, sketch the graph of the derivative $y = f'(x)$ , clearly showing the...
• 12N.1.hl.TZ0.4a: Find the coordinates of the points A and B.
• 12N.1.hl.TZ0.4b: Given that the graph of the function has exactly one point of inflexion, find its coordinates.
• 12N.2.hl.TZ0.12b: If a = 5 and b = 1, find the maximum length of a painting that can be removed through this doorway.
• 12N.2.hl.TZ0.12d: Let a = 3k and b = k .  Find, in terms of k , the maximum length of a painting that can be...
• 12N.2.hl.TZ0.12e: Let a = 3k and b = k .  Find the minimum value of k if a painting 8 metres long is to be removed...
• 08M.1.hl.TZ1.5: If $f(x) = x - 3{x^{\frac{2}{3}}},{\text{ }}x > 0$ , (a)     find the x-coordinate of the...
• 08M.1.hl.TZ1.12: The function f is defined by $f(x) = x{{\text{e}}^{2x}}$ . It can be shown that...
• 08M.2.hl.TZ1.13: A family of cubic functions is defined as...
• 08M.1.hl.TZ2.13: André wants to get from point A located in the sea to point Y located on a straight stretch of...
• 08M.2.hl.TZ2.6: Consider the curve with equation $f(x) = {{\text{e}}^{ - 2{x^2}}}{\text{ for }}x < 0$...
• 08N.1.hl.TZ0.9: A packaging company makes boxes for chocolates. An example of a box is shown below. This box is...
• 08N.2.hl.TZ0.12: The function f is defined by...
• 11M.1.hl.TZ2.1a: Find the value of p and the value of q .
• 11M.1.hl.TZ2.11a: Find the coordinates of the points on C at which $\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0$ .
• 09M.1.hl.TZ1.11: Let f be a function defined by $f(x) = x - \arctan x$ , $x \in \mathbb{R}$ . (a)     Find...
• 09N.1.hl.TZ0.9: The diagram below shows a sketch of the gradient function $f'(x)$ of the curve $f(x)$...
• SPNone.1.hl.TZ0.5c: John states that, because $f''(0) = 0$ , the graph of f has a point of inflexion at the point...
• SPNone.2.hl.TZ0.13b: Sketch the graphs of f and $f'$ on the same axes, showing clearly all x-intercepts.
• SPNone.2.hl.TZ0.13c: Find the x-coordinates of the two points of inflexion on the graph of f .
• 13M.2.hl.TZ1.7b: Find the value of x, to the nearest metre, such that this cost is minimized.
• 10M.1.hl.TZ1.11: Consider $f(x) = \frac{{{x^2} - 5x + 4}}{{{x^2} + 5x + 4}}$. (a)     Find the equations of all...
• 10M.1.hl.TZ2.7: The function f is defined by $f(x) = {{\text{e}}^{{x^2} - 2x - 1.5}}$. (a)     Find...
• 10M.2.hl.TZ2.11: The function f is defined...
• 10N.2.hl.TZ0.13: Let $f(x) = \frac{{a + b{{\text{e}}^x}}}{{a{{\text{e}}^x} + b}}$, where \(0 < b <...
• 13M.2.hl.TZ2.13e: Using the result in part (d), or otherwise, determine the value of x corresponding to the maximum...
• 11N.1.hl.TZ0.8c: What route should Jorg take to travel from A to B in the least amount of time? Give reasons for...
• 11N.1.hl.TZ0.11a: Determine the time at which the two ships are closest to one another, and justify your answer.
• 11N.1.hl.TZ0.11b: If the visibility at sea is 9 km, determine whether or not the captains of the two ships can ever...
• 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...
• 11M.2.hl.TZ1.2a: Find the equation of the straight line passing through the maximum and minimum points of the...
• 11M.2.hl.TZ1.2b: Show that the point of inflexion of the graph $y = f (x)$ lies on this straight line.  
• 09N.2.hl.TZ0.3: (a)     Show that the area of the shaded region is $8\sin x - 2x$ . (b)     Find the maximum...
• 09N.2.hl.TZ0.10: (a)     Find in terms of $a$   (i)     the zeros of $f$ ;   (ii)     the values of $x$...
• 09M.2.hl.TZ2.7: (a)     Show that ${b^2} > 24c$ . (b)     Given that the coordinates of P and Q are...
• 09M.2.hl.TZ2.12: (a)     Explain why $x < 40$ . (b)     Show that cosθ = x −10 50. (c)     (i)     Find an...
• 09M.2.hl.TZ2.13: (a)     On the same set of axes draw, on graph paper, the graphs, for...
• 14M.1.hl.TZ1.11c: Find the coordinates of C, the point of inflexion on the curve.
• 14M.1.hl.TZ2.13d: Find the $x$-coordinates of the other two points of inflexion.
• 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...
• 13N.1.hl.TZ0.10c: Find the coordinates of B, the point of inflexion.
• 14M.1.hl.TZ1.11b: Find the coordinates of B, at which the curve reaches its maximum value.
• 14M.1.hl.TZ2.13b: Hence find the $x$-coordinates of the points where the gradient of the graph of $f$ is zero.
• 13N.1.hl.TZ0.10b: Find an expression for $f''(x)$ and hence show the point A is a maximum.
• 14N.1.hl.TZ0.5: A tranquilizer is injected into a muscle from which it enters the bloodstream. The concentration...
• 14N.2.hl.TZ0.10c: (i)     Find $\frac{{{{\text{d}}^2}A}}{{{\text{d}}{x^2}}}$ and hence justify that...
• 15M.1.hl.TZ1.11c: Find the coordinates of any local maximum and minimum points on the graph of $y(x)$. Justify...
• 15M.1.hl.TZ1.11d: Find the coordinates of any points of inflexion on the graph of $y(x)$. Justify whether any...
• 15M.1.hl.TZ2.4b: There is a point of inflexion, $P$, on the curve $y = f(x)$. Find the coordinates of $P$.
• 15M.1.hl.TZ2.6b: Given that $AB$ has a minimum value, determine the value of $\theta$ for which this occurs.
• 15N.1.hl.TZ0.12c: Hence find the $x$-coordinates of any local maximum or minimum points.
• 15N.2.hl.TZ0.13b: Show that the gradient of the roof function is greatest when $x =  - \sqrt {200}$.
• 15N.2.hl.TZ0.13c: The cross section of the living space under the roof can be modelled by a rectangle $CDEF$ with...
• 16M.2.hl.TZ1.11b: For the curve $y = f(x)$. (i)     Find the coordinates of both local minimum points. (ii)    ...
• 16M.2.hl.TZ2.11c: (i)     Find $\frac{{\text{d}}}{{{\text{d}}x}}(\tan \alpha )$. (ii)     Hence or otherwise...
• 16M.2.hl.TZ2.11d: Find the set of values of $x$ for which $\alpha  \geqslant 7^\circ$.
• 16N.1.hl.TZ0.11c: Show that the function $f$ has a local maximum value when $x = \frac{{3\pi }}{4}$.
• 16N.1.hl.TZ0.11d: Find the $x$-coordinate of the point of inflexion of the graph of $f$.
• 16N.1.hl.TZ0.11e: Sketch the graph of $f$, clearly indicating the position of the local maximum point, the point...
• 17M.1.hl.TZ2.10a.i: Find the area of the window in terms of P and $r$.
• 17M.1.hl.TZ2.10a.ii: Find the width of the window in terms of P when the area is a maximum, justifying that this is a...
• 17M.1.hl.TZ2.10b: Show that in this case the height of the rectangle is equal to the radius of the semicircle.
• 18M.1.hl.TZ1.9a: The graph of $y = f\left( x \right)$ has a local maximum at A. Find the coordinates of A.
• 18M.1.hl.TZ1.9b.i: Show that there is exactly one point of inflexion, B, on the graph of $y = f\left( x \right)$.
• 18M.1.hl.TZ1.9b.ii: The coordinates of B can be expressed in the form...
• 18M.2.hl.TZ1.9a: Show that there are exactly two points on the curve where the gradient is zero.
• 18M.1.hl.TZ2.4: Consider the curve $y = \frac{1}{{1 - x}} + \frac{4}{{x - 4}}$. Find the x-coordinates of the...

6.4

• 09N.1.hl.TZ0.7b: Find $\int {{{\tan }^3}x{\text{d}}x}$ .
• 13N.2.hl.TZ0.10: By using the substitution $x = 2\tan u$, show that...
• 15M.1.hl.TZ1.3a: Find $\int {(1 + {{\tan }^2}x){\text{d}}x}$.
• 15M.1.hl.TZ1.3b: Find $\int {{{\sin }^2}x{\text{d}}x}$.
• 15M.1.hl.TZ1.6c: Hence, write down $\int {\frac{{3x - 2}}{{2x - 1}}} {\text{d}}x$.
• 18M.2.hl.TZ1.5a: Given that $2{x^3} - 3x + 1$ can be expressed in the...
• 18M.2.hl.TZ1.5b: Hence find $\int {\frac{{2{x^3} - 3x + 1}}{{{x^2} + 1}}} {\text{d}}x$.

6.5

• 12M.1.hl.TZ1.2b: Hence state the value of (i)     $f'( - 3)$; (ii)     $f'(2.7)$; (iii)  ...
• 12M.1.hl.TZ1.6a: Find the area of region A in terms of k .
• 12M.1.hl.TZ1.6b: Find the area of region B in terms of k .
• 12M.1.hl.TZ1.6c: Find the ratio of the area of region A to the area of region B .
• 12M.2.hl.TZ1.8: A cone has height h and base radius r . Deduce the formula for the volume of this cone by...
• 12M.1.hl.TZ2.10c: The region bounded by the graph, the x-axis and the y-axis is denoted by A and the region bounded...
• 12N.2.hl.TZ0.9: Find the area of the region enclosed by the curves $y = {x^3}$ and $x = {y^2} - 3$...
• 08M.1.hl.TZ1.6: Find the area between the curves $y = 2 + x - {x^2}{\text{ and }}y = 2 - 3x + {x^2}$ .
• 08M.1.hl.TZ1.10: The region bounded by the curve $y = \frac{{\ln (x)}}{x}$ and the lines x = 1, x = e, y = 0 is...
• 08M.2.hl.TZ2.3: The curve $y = {{\text{e}}^{ - x}} - x + 1$ intersects the x-axis at P. (a)     Find the...
• 08N.2.hl.TZ0.12: The function f is defined by...
• 11M.1.hl.TZ2.13a: (i)     Sketch the graphs of $y = \sin x$ and $y = \sin 2x$ , on the same set of axes, for...
• 11M.1.hl.TZ2.13c: The increasing function f satisfies $f(0) = 0$ and $f(a) = b$ , where $a > 0$ and...
• 09M.1.hl.TZ1.9: (a)     Let $a > 0$ . Draw the graph of $y = \left| {x - \frac{a}{2}} \right|$ for...
• 09M.1.hl.TZ2.4: (a)     Show that \(\frac{3}{{x + 1}} + \frac{2}{{x + 3}} = \frac{{5x + 11}}{{{x^2} + 4x +...
• 09M.1.hl.TZ2.5: Consider the part of the curve $4{x^2} + {y^2} = 4$ shown in the diagram below.     (a)  ...
• 09M.1.hl.TZ2.11: A function is defined as $f(x) = k\sqrt x$, with $k > 0$ and $x \geqslant 0$ . (a)    ...
• 09N.1.hl.TZ0.10: A drinking glass is modelled by rotating the graph of $y = {{\text{e}}^x}$ about the y-axis,...
• SPNone.3ca.hl.TZ0.4b: Determine the value of $\int_{ - a}^a {f(x){\text{d}}x}$ where $a > 0$ .
• 13M.1.hl.TZ1.10b: Find \(\int_{\frac{1}{{n + 1}}}^{\frac{1}{n}} {\pi {x^{ - 2}}\sin (\pi {x^{ - 1}}){\text{d}}x}...
• 13M.1.hl.TZ1.10c: Evaluate $\int_{0.1}^1 {\left| {\pi {x^{ - 2}}\sin (\pi {x^{ - 1}})} \right|{\text{d}}x}$.
• 13M.2.hl.TZ1.4: Find the volume of the solid formed when the region bounded by the graph of $y = \sin (x - 1)$,...
• 10M.1.hl.TZ1.8: The region enclosed between the curves $y = \sqrt x {{\text{e}}^x}$ and...
• 10M.2.hl.TZ1.10: The diagram below shows the graphs of $y = \left| {\frac{3}{2}x - 3} \right|,{\text{ }}y = 3$...
• 10M.2.hl.TZ2.11: The function f is defined...
• 10N.1.hl.TZ0.13: Consider the curve $y = x{{\text{e}}^x}$ and the line \(y = kx,{\text{ }}k \in...
• 10N.2.hl.TZ0.13: Let $f(x) = \frac{{a + b{{\text{e}}^x}}}{{a{{\text{e}}^x} + b}}$, where \(0 < b <...
• 13M.1.hl.TZ2.1: Find the exact value of...
• 13M.2.hl.TZ2.12b: A different solution of the differential equation, satisfying y = 2 when $x = \frac{\pi }{4}$,...
• 11N.1.hl.TZ0.4c: find the volume of the solid formed when the graph of f is rotated through $2\pi$ radians...
• 11N.1.hl.TZ0.7: The graphs of $f(x) = - {x^2} + 2$ and $g(x) = {x^3} - {x^2} - bx + 2,{\text{ }}b > 0$,...
• 11N.2.hl.TZ0.1b: Find the area of the region bounded by the graph and the x and y axes.
• 11M.1.hl.TZ1.7: Find the area enclosed by the curve $y = \arctan x$ , the x-axis and the line $x = \sqrt 3$ .
• 11M.2.hl.TZ1.14a: When the glass contains water to a height $h$ cm, find the volume $V$ of water in terms of...
• 09M.2.hl.TZ1.12: (a)     If A, B and C have x-coordinates $a\frac{\pi }{2}$, $b\frac{\pi }{6}$ and...
• 14M.1.hl.TZ1.5c: Hence find the value of...
• 14M.1.hl.TZ1.6: The first set of axes below shows the graph of $y = {\text{ }}f(x)$ for...
• 14M.1.hl.TZ1.11e: The graph of $y = {\text{ }}f(x)$ crosses the $x$-axis at the point A. Find the area...
• 14M.2.hl.TZ1.5b: The region $S$ is rotated by $2\pi$ about the $x$-axis to generate a solid. (i)     Write...
• 14M.1.hl.TZ2.13e: Find the area of the shaded region. Express your answer in the form...
• 14M.2.hl.TZ2.3b: Find the area enclosed between the two graphs for...
• 13N.1.hl.TZ0.10f: Find an exact value for the area of the region bounded by the curve $y = g(x)$, the x-axis and...
• 13N.1.hl.TZ0.12f: S is rotated through $2\pi$ radians about the x-axis. Find the value of the volume generated.
• 13N.2.hl.TZ0.13b: The domain of $f$ is now restricted to $x \geqslant 0$. (i)     Find an expression for...
• 13N.1.hl.TZ0.12e: Hence find the value of $\int_0^{\frac{\pi }{2}} {{{\cos }^6}\theta {\text{d}}\theta }$.
• 14N.1.hl.TZ0.11d: A region $R$ is bounded by the graphs of $y = g(x)$, the tangent $T$ and the line...
• 14N.2.hl.TZ0.13a: If the container is filled with water to a depth of $h\,{\text{cm}}$, show that the volume,...
• 15M.1.hl.TZ2.11d: Show that the area bounded by the graph of $y = g \circ f(x)$, the $x$-axis and the lines...
• 15M.2.hl.TZ1.1: The region $R$ is enclosed by the graph of $y = {e^{ - {x^2}}}$, the $x$-axis and the lines...
• 15M.2.hl.TZ2.6b: The region bounded by the graph of $y = \ln (5x + 10)$, the $x$-axis and the lines...
• 15N.1.hl.TZ0.12f: Find the area of the region enclosed by the graph of $y = f(x)$ and the $x$-axis for...
• 15N.1.hl.TZ0.12g: Show that...
• 15N.2.hl.TZ0.5a: (i)     Express the area of the region $R$ as an integral with respect to $y$. (ii)    ...
• 15N.2.hl.TZ0.5b: Find the exact volume generated when the region $R$ is rotated through $2\pi$ radians about...
• 16M.1.hl.TZ2.3b: Hence find...
• 16M.1.hl.TZ2.11a: Calculate the value of the volume generated.
• 16M.1.hl.TZ1.13d: Find the volume of the solid formed when the region $R$ is rotated through $2\pi$ about the...
• 16M.2.hl.TZ2.12a: (i)     Show that...
• 16N.1.hl.TZ0.11f: Find the area of the region enclosed by the graph of $f$ and the $x$-axis. The curvature at...
• 16N.2.hl.TZ0.10f: Consider the region $R$ enclosed by the graph of $y = f(x)$ and the axes. Find the volume of...
• 17M.1.hl.TZ1.11d: Hence find the value of $p$ if $\int_0^1 {f(x){\text{d}}x = \ln (p)}$.
• 17M.1.hl.TZ1.11f: Determine the area of the region enclosed between the graph of...
• 17M.2.hl.TZ1.4a: Write down a definite integral to represent the area of $A$.
• 17M.2.hl.TZ1.4b: Calculate the area of $A$.
• 17M.2.hl.TZ2.2b: Find the volume of the solid formed when the region bounded by the curve, the $x$-axis for...
• 17N.2.hl.TZ0.10d: This region is now rotated through $2\pi$ radians about the $x$-axis. Find the volume of...
• 18M.1.hl.TZ1.4a: $\int_{ - 2}^0 {\left( {f\left( x \right){\text{ + 2}}} \right){\text{d}}x}$.
• 18M.1.hl.TZ1.4b: $\int_{ - 2}^0 {f\left( {x{\text{ + 2}}} \right){\text{d}}x}$.
• 18M.1.hl.TZ1.7b: Find $\int_0^1 {{\text{arccos}}\left( {\frac{x}{2}} \right){\text{d}}x}$.
• 18M.2.hl.TZ1.9d: The shaded region is rotated by 2$\pi$ about the $y$-axis. Find the volume of the solid formed.
• 18M.1.hl.TZ2.6b: Hence, or otherwise, find...
• 18M.1.hl.TZ2.11c: The region R, is bounded by the graph of the function found in part (b), the x-axis, and...

6.6

• 12M.2.hl.TZ2.12c: (i)     Write down the time T at which the velocity is zero. (ii)     Find the distance...
• 12M.2.hl.TZ2.12d: Find an expression for s , the displacement, in terms of t , given that s = 0 when t = 0 .
• 12M.2.hl.TZ2.12e: Hence, or otherwise, show that $s = \frac{1}{2}\ln \frac{2}{{1 + {v^2}}}$.
• 12N.2.hl.TZ0.6: A particle moves along a straight line so that after t seconds its displacement s , in...
• 08M.2.hl.TZ2.13: A particle moves in a straight line in a positive direction from a fixed point O. The velocity v...
• 11M.2.hl.TZ2.3a: Find her acceleration 10 seconds after jumping.
• 11M.2.hl.TZ2.3b: How far above the ground is she 10 seconds after jumping?
• SPNone.2.hl.TZ0.5a: Given that P is at the origin O at time t = 0 , calculate (i)     the displacement of P from O...
• SPNone.2.hl.TZ0.5b: Find the time at which the total distance travelled by P is 1 m.
• 13M.2.hl.TZ1.12d: At t = 0 the particle is at point O on the line. Find an expression for the particle’s...
• 13M.2.hl.TZ1.12e: A second particle, B, moving along the same line, has position ${x_B}{\text{ m}}$, velocity...
• 13M.2.hl.TZ1.12f: Find the value of t when the two particles meet.
• 10M.2.hl.TZ1.14: A body is moving through a liquid so that its acceleration can be expressed...
• 10N.1.hl.TZ0.12a: A particle P moves in a straight line with displacement relative to origin given...
• 13M.2.hl.TZ2.10: The acceleration of a car is $\frac{1}{{40}}(60 - v){\text{ m}}{{\text{s}}^{ - 2}}$, when its...
• 11M.2.hl.TZ1.8a: Find an expression for the acceleration of the jet plane during this time, in terms of $t$ .
• 11M.2.hl.TZ1.8b: Given that when $t = 0$ the jet plane is travelling at $125$ ms−1, find its maximum velocity...
• 11M.2.hl.TZ1.8c: Given that the jet plane breaks the sound barrier at $295$ ms−1, find out for how long the jet...
• 14M.1.hl.TZ1.8: A body is moving in a straight line. When it is $s$ metres from a fixed point O on the line its...
• 14M.2.hl.TZ2.14d: Find the acceleration of particle B when $s = 0.1{\text{ m}}$.
• 14M.2.hl.TZ2.14c: Find the exact distance travelled by particle $A$ between $t = 0$ and $t = 6$...
• 14N.2.hl.TZ0.8a: Find the value of $t$ when the particle is instantaneously at rest.
• 14N.2.hl.TZ0.8b: The particle returns to its initial position at $t = T$. Find the value of T.
• 15M.2.hl.TZ1.13a: (i)     Find his acceleration $a(t)$ for $t < 10$. (ii)     Calculate $v(10)$. (iii)  ...
• 15M.2.hl.TZ1.13b: At $t = 10$ his parachute opens and his acceleration $a(t)$ is subsequently given by...
• 15M.2.hl.TZ1.13c: You are told that Richard’s acceleration, $a(t) =  - 10 - 5v$, is always positive, for...
• 15M.2.hl.TZ1.13d: You are told that Richard’s acceleration, $a(t) =  - 10 - 5v$, is always positive, for...
• 15M.2.hl.TZ1.13e: You are told that Richard’s acceleration, $a(t) =  - 10 - 5v$, is always positive, for...
• 15M.2.hl.TZ1.13f: You are told that Richard’s acceleration, $a(t) =  - 10 - 5v$, is always positive, for...
• 15M.2.hl.TZ2.12a: Find the displacement of the particle when $t = 4$.
• 15M.2.hl.TZ2.12b: Sketch a displacement/time graph for the particle, $0 \le t \le 5$, showing clearly where the...
• 15M.2.hl.TZ2.12c: For $t > 5$, the displacement of the particle is given by...
• 15M.2.hl.TZ2.12d: For $t > 5$, the displacement of the particle is given by...
• 15N.2.hl.TZ0.9a: Write down the first two times ${t_1},{\text{ }}{t_2} > 0$, when the particle changes...
• 15N.2.hl.TZ0.9b: (i)     Find the time $t < {t_2}$ when the particle has a maximum velocity. (ii)     Find...
• 15N.2.hl.TZ0.9c: Find the distance travelled by the particle between times $t = {t_1}$ and $t = {t_2}$.
• 16M.2.hl.TZ1.3a: Find an expression for the velocity, $v$, of the particle at time $t$.
• 16M.2.hl.TZ1.3b: Find an expression for the acceleration, $a$, of the particle at time $t$.
• 16M.2.hl.TZ1.3c: Find the acceleration of the particle at time $t = 0$.
• 16M.2.hl.TZ2.8a: Find the particle’s acceleration in terms of $s$.
• 16M.2.hl.TZ2.8b: Using an appropriate sketch graph, find the particle’s displacement when its acceleration is...
• 17M.1.hl.TZ2.4a: Find ${t_1}$ and ${t_2}$.
• 17M.1.hl.TZ2.4b: Find the displacement of the particle when $t = {t_1}$
• 17M.2.hl.TZ1.11a: Find his velocity when $t = 15$.
• 17M.2.hl.TZ1.11b: Calculate the vertical distance Xavier travelled in the first 10 seconds.
• 17M.2.hl.TZ1.11c: Determine the value of $h$.
• 17N.1.hl.TZ0.5: A particle moves in a straight line such that at time $t$ seconds $(t \geqslant 0)$, its...
• 18M.2.hl.TZ2.7a: Determine the first time t1 at which P has zero velocity.
• 18M.2.hl.TZ2.7b.i: Find an expression for the acceleration of P at time t.
• 18M.2.hl.TZ2.7b.ii: Find the value of the acceleration of P at time t1.

6.7

• 12M.1.hl.TZ1.12c:  Use the substitution $x = {\sin ^2}\theta$ to show that...
• 12M.1.hl.TZ2.10c: The region bounded by the graph, the x-axis and the y-axis is denoted by A and the region bounded...
• 12N.2.hl.TZ0.8: By using the substitution $x = \sin t$ , find...
• 08M.2.hl.TZ1.9: By using an appropriate substitution...
• 08M.1.hl.TZ2.6: Show that...
• 08N.1.hl.TZ0.5: Calculate the exact value of $\int_1^{\text{e}} {{x^2}\ln x{\text{d}}x}$ .
• 11M.1.hl.TZ2.13b: Find the value of $\int_0^1 {\sqrt {\frac{x}{{4 - x}}} }{{\text{d}}x}$ using the substitution...
• 11M.2.hl.TZ2.13B: (a)     Using integration by parts, show that...
• 11M.3ca.hl.TZ0.4a: Show that ${I_0} = \frac{1}{2}(1 + {{\text{e}}^{ - \pi }})$ .
• 11M.3ca.hl.TZ0.4b: By letting $y = x - n\pi$ , show that ${I_n} = {{\text{e}}^{ - n\pi }}{I_0}$ .
• 09N.1.hl.TZ0.7a: Calculate...
• SPNone.1.hl.TZ0.11a: Find the value of the integral $\int_0^{\sqrt 2 } {\sqrt {4 - {x^2}} {\text{d}}x}$ .
• SPNone.1.hl.TZ0.11b: Find the value of the integral $\int_0^{0.5} {\arcsin x {\text{d}}x}$ .
• SPNone.1.hl.TZ0.11c: Using the substitution $t = \tan \theta$ , find the value of the...
• 13M.1.hl.TZ1.12e: By using a suitable substitution show that...
• 10M.1.hl.TZ1.9: (a)     Given that $\alpha  > 1$, use the substitution $u = \frac{1}{x}$ to show...
• 10M.1.hl.TZ2.9: Find the value of $\int_0^1 {t\ln (t + 1){\text{d}}t}$.
• 13M.2.hl.TZ2.4a: Find $\int {x{{\sec }^2}x{\text{d}}x}$.
• 09M.2.hl.TZ1.6: (a)     Integrate $\int {\frac{{\sin \theta }}{{1 - \cos \theta }}} {\text{d}}\theta$...
• 09M.2.hl.TZ2.9: Using the substitution $x = 2\sin \theta$ , show...
• 14M.2.hl.TZ1.6b: Find $\int {f(x){\text{d}}x}$.
• 14M.1.hl.TZ2.10: Use the substitution $x = a\sec \theta$ to show that...
• 14M.2.hl.TZ2.14b: Use the substitution $u = {t^2}$ to find $\int {\frac{t}{{12 + {t^4}}}{\text{d}}t}$.
• 14N.1.hl.TZ0.6: By using the substitution $u = 1 + \sqrt x$, find...
• 15M.1.hl.TZ1.8: By using the substitution $u = {{\text{e}}^x} + 3$, find...
• 15M.1.hl.TZ2.5: Show that $\int_1^2 {{x^3}\ln x{\text{d}}x = 4\ln 2 - \frac{{15}}{{16}}}$.
• 15M.1.hl.TZ2.8: By using the substitution $t = \tan x$, find...
• 15N.1.hl.TZ0.2: Using integration by parts find $\int {x\sin x{\text{d}}x}$.
• 15N.1.hl.TZ0.5: Use the substitution $u = \ln x$ to find the value of...
• 15N.1.hl.TZ0.12f: Find the area of the region enclosed by the graph of $y = f(x)$ and the $x$-axis for...
• 16M.1.hl.TZ1.13b: Use the substitution $u = \ln x$ to find the area of the region $R$.
• 16M.1.hl.TZ1.13c: (i)     Find the value of ${I_0}$. (ii)     Prove that...
• 16N.1.hl.TZ0.11f: Find the area of the region enclosed by the graph of $f$ and the $x$-axis. The curvature at...
• 17M.1.hl.TZ1.9: Find $\int {\arcsin x\,{\text{d}}x}$
• 17M.1.hl.TZ2.6a: Using the substitution $x = \tan \theta$ show that...
• 17M.1.hl.TZ2.6b: Hence find the value of...
• 17M.1.hl.TZ2.9b: Find $\int {f(x)\cos x{\text{d}}x}$.
• 17N.2.hl.TZ0.8: By using the substitution ${x^2} = 2\sec \theta$, show that...
• 18M.1.hl.TZ2.6b: Hence, or otherwise, find...
• 18M.1.hl.TZ2.8a: Use the substitution $u = {x^{\frac{1}{2}}}$ to...
• 18M.1.hl.TZ2.8b: Hence find the value...