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...