Free Printable Calculus

1.

Find the gradient of the curve with equation

2x² − 4xy + 3y² = 3, at the point (2, 1).

a)

2

b)

3

c)

4

d)

5

2.

The equation of a curve is x²y + y² = 6x

the equation of the tangent to the curve at the point with coordinates (1, 2)

a)

3x – 5y + 8 = 0

b)

2x – 5y + 8 = 0

c)

2x – 5y - 8 = 0

d)

3x – 5y - 8 = 0

3.

A solid rectangular block has a square base of side x cm. The height of the block is h cm and the total surface area of the block is 96 cm^2.Find the stationary value of V

a)

V = 84

b)

V = 74

c)

V = 64

d)

V = 54

4.

The diagram shows the curve y = (x − 2)² and the line y + 2x = 7, which intersect at points A and B. Find the area of the shaded region.

a)

$10\ \frac{5}{7}$

b)

$10\ \frac{3}{4\ }$

c)

$10\ \frac{1}{3\ }$

d)

$10\ \frac{2}{3}$

5.

$\int_{ }^{ }\left(x^{3\ }+\frac{1}{x^3}\right)\ dx$

a)

$\frac{x^{4\ }}{4}-\frac{x^{-2}}{2\ }+c$

b)

$\frac{x^{4\ }}{4}+\frac{x^{-2}}{2}+c$

c)

$\frac{x^4}{4}-\frac{x^2}{2\ }+c$

d)

$\frac{-x^4}{4}+\frac{x^{2\ }}{2\ }+c$

6.

the equation of the curve $y=\frac{4}{3x-4}$
and P(2, 2) is a point on the curve. The equation of the tangent to the curve at P and the angle that this tangent makes with the x-axis respectively

a)

y − 2 = −3(x − 2) and $108.4^o$

b)

y − 2 = 3(x − 2) and $128.8^o$

c)

y − 2 = −3(x + 2) and $78.3^o$

d)

y + 2 = −3(x − 2) and $45^o$

7.

The diagram shows part of the curve $y=4\sqrt{x}-x$
The curve has a maximum point at M and meets the x-axis at O and A. Find the volume obtained when the shaded region is rotated through 360o about the x-axis, givingyour answer in terms of π

a)

138.5π

b)

137.5π

c)

136.5π

d)

135.5π

8.

The equation of a curve is $y=4\sqrt{x}+\frac{2}{\sqrt{x}}$
A point is moving along the curve in such a way that the x-coordinate is increasing at a constant rate of 0.12 units per second. Find the rate of change of the y-coordinate when x = 4

a)

0.105

b)

0.125

c)

0.345

d)

0.546

9.

The diagram shows part of the curve y = −x² + 8x − 10 which passes through the points A and B. The curve has a maximum point at A and the gradient of the line BA is 2. evaluate the area of the shaded region.

a)

$9\ \frac{1}{3}$

b)

$10\ \frac{1}{3}$

c)

$11\ \frac{1}{3}$

d)

$12\ \frac{1}{3}$

10.

A curve is such that $\frac{dy}{dx}=\frac{6}{x^2}$ and (􏰎2, 9)􏰏 is a point on the curve. Find the equation of the curve

a)

$y=-6x^{-1}+12$

b)

$y=6x^{-1}-12$

c)

$y=-6x^{-2}+12$

d)

$y=6x^{-2}+12$

11.

The straight line y = mx + 14 is a tangent to the curve $y=\frac{12}{x}+2$
at the point P. Find the value of the constant m and the coordinates of P

a)

m = 3 and y = - 8

b)

m = −3 and y = 8

c)

m = 8 and y = - 3

d)

m = −8 and y = 3

12.

The diagram shows the curve $y=\sqrt{1+4x}$
which intersects the x-axis at A and the y-axis at B. The normal to the curve at B meets the x-axis at C. the area of the shaded region

a)

$\frac{6}{5}$

b)

$\frac{5}{6}$

c)

$\frac{6}{7}$

d)

$\frac{7}{6}$

Calculus

Total questions: 12

Worksheet time: 3600secs

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