DP Mathematics HL Questionbank

• Syllabus

6.1

Sub sections and their related questions

Informal ideas of limit, continuity and convergence.

• 14M.1.hl.TZ2.8a: Determine whether or not \(f\)is continuous.
• 15M.2.hl.TZ2.12c: For \(t > 5\), the displacement of the particle is given by...
• 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.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.TZ0.11g: 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...
• 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...

Definition of derivative from first principles as \(f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} {\frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}} \).

• 12M.1.hl.TZ2.13a: Using the definition of a derivative as...
• SPNone.1.hl.TZ0.13a: Given that f and its derivative, \(f'\) , are continuous for all values in the domain of f , find...
• 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.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.TZ0.11g: 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...
• 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...

The derivative interpreted as a gradient function and as a rate of change.

• 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.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 .
• 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.
• 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.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.TZ0.11g: 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...
• 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...

Finding equations of tangents and normals.

• 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.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.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.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.TZ0.13: 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.TZ0.13a: 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.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.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...
• 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 \((6,{\text{ }}1)\).
• 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...
• 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.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.TZ0.11g: 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...
• 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...

Identifying increasing and decreasing 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...
• 14M.3ca.hl.TZ0.2a: Consider the functions \(f(x) = {(\ln x)^2},{\text{ }}x > 1\) and...
• 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.
• 17M.1.hl.TZ2.9c: By finding \(g'(x)\) explain why \(g\) is an increasing function.
• 17M.2.hl.TZ2.2a: Find the equation of the normal to the curve at the point \(\left( {1,{\text{ }}\sqrt 3 } \right)\).

The second derivative.

• SPNone.1.hl.TZ0.5b: Show that \(f''(0) = 0\) .
• 15N.3ca.hl.TZ0.2a: Show that \(f''(x) = 2\left( {f'(x) - f(x)} \right)\).
• 15N.2.hl.TZ0.13a: Find \(f''(x)\).
• 17M.1.hl.TZ2.9c: By finding \(g'(x)\) explain why \(g\) is an increasing function.
• 17M.2.hl.TZ2.2a: Find the equation of the normal to the curve at the point \(\left( {1,{\text{ }}\sqrt 3 } \right)\).

Higher derivatives.

• 10N.1.hl.TZ0.12b: Consider the...
• 11N.1.hl.TZ0.6: Given that \(y = \frac{1}{{1 - x}}\), use mathematical induction to prove that...
• 15N.1.hl.TZ0.8b: Consider \(f(x) = \sin (ax)\) where \(a\) is a constant. Prove by mathematical induction that...
• 16M.1.hl.TZ2.11c: (i)     Find \(\frac{{{{\text{d}}^2}h}}{{{\text{d}}{t^2}}}\). (ii)     Find the values of...
• 16N.1.hl.TZ0.11b: Show that \(\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}} = 2{{\text{e}}^x}\cos x\).
• 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...