On this page you can find examination questions from the topic of calculus
Find the value(s) of x for which the graph \(y=x^3-8x+2\) has gradient 4
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Find \(f'(4)\) for the function \(f(x)=2x+\frac { 8 }{ \sqrt { x } } +\frac { 32 }{ x } \)
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The graph of y = f(x) is shown below, where B is a local maximum and C is a local minimum
Sketch a graph of y = f'(x), clearly showing the images of the points B and C labellling them B' and C' respectively
Hint
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A function is given by \(f(x)=-x^3+6x^2+4\)
a) Find the coordinates of any stationary points and describe their nature
b) Determine the values of x such that f(x) is a increasing function
c) Find the coordinates of the point of inflexion
Hint
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The following diagram shows the graph of \(f'\), the derivative of f
On the graph below, sketch the graph of y = f(x) given that f(0) = 0. Mark the images of A , B and C labelling them A' , B' and C'.
Hint
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Consider the function \(f(x)=-x^3-3x^2+9x\)
a) Find the coordinates of any stationary points and determine their nature
b) Find the equation of the straight line that passes through both the local maximum and the local minimum points.
c) Show that the point of inflexion lies on this line.
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Let \(y=xe^x\)
a) Find \(\frac{dy}{dx}\)
b) Show that \(\frac{d^2y}{dx^2}=e^x(2+x)\)
c) Find the coordinates of the stationary point and show that it is a local minimum.
Hint
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Let \(f(x)={ x }^{ 2 }{ (2x-3) }^{ 3 }\)
a) Find \(f'(x)\)
b) The graph of y = f(x) has stationary points at x = 0, x= \(\frac{3}{2}\) and x = a. Find the value of a
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Let \(f(x)=\frac{lnx}{x},x>0\)
a) Show that \(f'(x)=\frac{1-lnx}{x^2}\)
b) Find \(f''(x)\)
c) The graph of f has a point of inflexion at A. Find the x-coordinate of A.
Hint
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Let \(f(x)=tanx\). A\((\frac{\pi}{3},\sqrt{3})\) is a point that lies on the graph of \(y=f(x)\)
a) Given that \(tanx=\frac{sinx}{cosx}\) find \(f'(x)\)
b) Show that \(f'(\frac{\pi}{3})=4\)
c) Find the equation of the normal to the curve y = f(x) at the point A
d) Show that the normal crosses the y axis at \(\sqrt{3}+\frac{\pi}{12}\)
Hint
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Let f(x) = (x - 1)(x - 4)(x + 2). The diagram below shows the graph of f and the point P where the graph crosses the x axis.
The line L is the tangent to the graph of f at the point P.
The line L intersects the graph of f at another point Q, as shown below
a) Find the coordinates of P
b) Show that \(f(x)=x^3-3x^2-6x+8\)
c) Find the equation of L in the form y = ax + b
d) Find the x coordinate of Q.
Hint
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Let \(f(x)=\frac{x^4-4x^2}{4}\) .
C(2 , 0) lies on the graph of y = f(x)
a) The tangent to the graph of y = f(x) at C cuts the y axis at A. Find the coordinates of A.
b) The normal to the graph of y = f(x) at C cuts the y axis at B. Find the area of the triangle ABC.
Hint
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The function \(f(x)=x^3-x^2-9x+9\) intersects the x axis at A, B and C.
The x coordinate of the point D is the mean of the x coordinates of B and C.
a) Find the coordinates of A, B and C.
b) Find the equation of the tangent to the curve at D.
c) Find the point of the intersection of the tangent with the curve. Interpret your result.
Hint
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A container is made from a cylinder and a hemisphere. The radius of the cylinder is r m and the height is h. The volume of the container is \(45\pi\)
a) Find an expression for the height of the cylinder in terms of r
b) Show that the surface area of the container, \(A=\frac{5 \pi r^2}{3}+\frac{90 \pi}{r}\)
c) Hence, find the values of r and h that give the minimum surface area of the container
* Volume of a sphere = \(\frac{4}{3}\pi r^3\)
** Surface area of a sphere = \(4\pi r^2\)
Hint
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The diagram below shows the graph of the functions f(x) = sinx and g(x) = 2sinx
A rectangle ABCD is placed in between the two functions as shown so that B and C lie on g , BC is parallel to the x axis and the local minima of the function f lies on AD.
Let NA = x
a) Find an expression for the height of the rectangle AB
b) Show that the area of the rectangle, A can be given by A = 4xcosx - 2x
c) Find \(\frac{dA}{dx}\)
d) Find the maximum value of the area of the rectangle.
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Given that \(\int _{ 4 }^{ 8 }{ \frac { 1 }{ 2x-4 } dx= } ln\sqrt { a } \) , find the value of a
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Consider a function f(x) such that \(\int _{ 0 }^{ 4 }{ f(x)dx } \) = 6
Find
a) \(\int _{ 0 }^{ 4 }{3 f(x)dx } \)
b) \(\int _{ 0 }^{ 4 }{[ f(x)+3]dx } \)
c) \(\int _{ -3 }^{ 1 }{\frac{1}{3} f(x+3)dx } \)
d) \(\int _{ 0 }^{ 4 }{ [f(x)+x]dx } \)
Hint
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Let f(x)=sinx, for \(0\le x\le 2\pi \)
The following diagram shows the graph of f
The shaded region R is enclosed by the graph of f, the line x=a , where a<\(\pi\) and the x-axis.
The area of R is \(\left( 1-\frac { \sqrt { 3 } }{ 2 } \right) \). Find the value of b.
Hint
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Show that the area bounded by the graphs of y = f(x) and y = g(x) in the interval \(0\le x\le \pi \) is given by 2 - \(\frac { \pi }{ 2} \)
f(x) = sinx
g(x) = sin²x
Hint
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a) Find \(\int { \frac { { e }^{ x } }{ 1+{ e }^{ x } } dx } \)
b) Evaluate \(\int _{ \frac { { \pi }^{ 2 } }{ 4 } }^{ \pi ^{ 2 } }{ \frac { cos\sqrt { x } }{ \sqrt { x } } } dx\)
Hint
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Video Solution
The following diagram shows the graph of f(x) = \(\frac { 4x }{ \sqrt { { x }^{ 2 }+1 } }\)
Let R be the region bounded by f, the x-axis, x = 1 and x = 2
Find R
Hint
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a) Using the fact that \(tanx = \frac{sinx}{cosx} \), show that \(\frac{d}{dx}(tanx) = \frac{1}{cos^2x}\)
b) Hence, find \(\int { \frac { \sqrt { tanx } }{ cos^{ 2 }x } } dx\)
Hint
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Video Solution
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