Calculus Revision Assessment 1

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Calculus Revision Assessment 1 - Calculus

Welcome to the Calculus Revision Assessment 1. There are 22 questions in total in different topics. You will need your calculator at hand.
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1 of 22

Below is a list of functions that corresponds to the statement, f(g(h(j(1))))=1.

$f (x) = a x^{} + 3$

$g (x) = x^{} - 2$

$h (x) = x^{} - 2$

$j (x) = x^{} + 2$

As you can see we have removed one of the coefficients in the list of functions and replaced it with a.

Use the information that has been given to you and your own knowledge of function composition to find the correct value of a correct to 2 decimal places if need be.

Input your answer in the box below

2 of 22

Enrico was going to fill the water in the tank as shaped below:

The water has been filled at a constant rate of 60 lit/s
Which of the following will represent the rate of change of height of water?

$0.022 m / s$

$0.022 cm / s$

$22.222 cm / s$

$0.002 cm / s$
None of these!
I don't know!

3 of 22

What is the derivative with respect to x of the following expression?

$f (x) = {(2 - 6 x)}^{3}$

$- 12 + 72 x - 108 x^{2}$

$- 72 + 432 x - 108 x^{2}$

$- 72 + 216 x - 648 x^{2}$

$- 72 + 216 x - 216 x^{2}$

None of these

I don't know

4 of 22

What is the derivative with respect to x of the following expression?

$ln (4 x^{8} - sin (6 x))$

$\frac{4 x^{7} - 6 cos (6 x)}{4 x^{8} - sin (6 x)}$

$\frac{8 x^{7} - 6 cos (6 x)}{4 x^{8} - sin (6 x)}$

$\frac{32 x^{7} + 6 cos (6 x)}{4 x^{8} - sin 6 x}$

$\frac{32 x^{7} - 6 cos (6 x)}{4 x^{8} - sin (6 x)}$

None of these

I don't know

5 of 22

Differentiate

$y = 3 x^{sin 2 x}$
with respect to x.

$\frac{d y}{d x} = 2 cos (2 x) ln (3 x) + \frac{1}{x} sin (2 x)$

$\frac{d y}{d x} = (2 cos (2 x) ln (x) + \frac{1}{x} sin (2 x)) 3 x^{sin 2 x}$

$\frac{d y}{d x} = 2 cos (2 x) ln (3 x) + \frac{1}{3 x} sin (2 x)$

$\frac{d y}{d x} = (2 cos (2 x) ln (3 x) + \frac{1}{3 x} sin (2 x)) 3 x^{sin (2 x)}$

None of these!

I don't know!

6 of 22

Differentiate

$(7 + 3 x + 2 x^{2}) e^{- 11 x}$ with respect to x.
The options below may not be fully simplified.

$- 11 (7 + 3 x + 2 x^{2}) e^{- 11 x} + (3 + x^{}) e^{- 11 x}$

$- 11 (7 + 3 x + 2 x^{2}) e^{- 11 x} + (3 + 4 x) e^{- 11 x}$

$- 11 (7 + 3 x + 2 x^{2}) e^{- 11 x} + (3 + 2 x) e^{- 11 x}$

$- 11 (7 + 3 x + 2 x^{2}) e^{- 11 x} + (- 11 - 16 x) e^{- 11 x}$

None of these!

I don't know!

7 of 22

Differentiate

$\frac{- 3 + 7 x}{1 + 4 x}$

with respect to x.

$\frac{3}{{(1 + 4 x)}^{1 / 2}}$

$\frac{19}{{(1 + 4 x)}^{2}}$

$\frac{- 5 + 56 x}{{(1 + 4 x)}^{2}}$

$\frac{- 2 + 11 x}{{(1 + 4 x)}^{2}}$

None of these!

I don't know!

8 of 22

Differentiate

${(4 x^{2} + 12 x^{3} + 13 x^{4})}^{4}$

with respect to x.

${4 (4 x^{2} + 12 x^{3} + 13 x^{4})}^{3} (8 x + 36 x^{2} + 52 x^{3})$

${4 (4 x^{2} + 12 x^{3} + 13 x^{4})}^{3} {(4 x + 12 x^{2} + 13 x^{3})}^{4}$

${4 (4 x^{2} + 12 x^{3} + 13 x^{4})}^{3} {(12 x^{2} + 26 x^{3})}^{4}$

${4 (4 x^{2} + 12 x^{3} + 13 x^{4})}^{3} {(2 x^{} + 4 x^{2} + \frac{13}{4} x^{3})}^{4}$

None of these!

I don't know!

9 of 22

What is the derivative with respect to x of the following expression?

${(11 x - cot (9 x))}^{4}$

$(44 - 36 cosec (9 x)) {(11 x - cot (9 x))}^{3}$

$(44 + 36 cosec (9 x)) {(11 x - cot (9 x))}^{3}$

$(44 + 36 {cosec}^{2} (9 x)) {(11 x - cot (9 x))}^{3}$

$(44 - 36 {cosec}^{2} (9 x)) {(11 x - cot (9 x))}^{3}$

None of these

I don't know

10 of 22

The function:

$y = - 3 x^{2} + 9 x^{} - 9$
with the domain:

$[4, 7]$
has the RANGE:

$[p, q]$
Find the values of p and q.

p =

q =

Provide your answer to 2 decimal places if necessary. Remember to type the values in the correct order, and to include minus signs where necessary.

11 of 22

What is the derivative with respect to x of the following expression?

$f (x) = -8 e^{7 x} + 5 ln (4 x)$

$-56 e^{7 x} - \frac{5}{4 x}$

$-8 e^{7 x} - \frac{5}{x}$

$-8 e^{7 x} - \frac{5}{4 x}$

$-56 e^{7 x} + \frac{5}{4 x}$

None of these

I don't know

12 of 22

What is the derivative with respect to x of the following expression?
The options below may not be fully simplified.

$e^{5 x^{6}} \sin (5 x^{10} + 6)$

$e^{5 x^{6}} 50 x^{9} \cos (5 x^{10} + 6) - 30 x^{5} e^{5 x^{6}} \sin (5 x^{10} + 6)$

$e^{5 x^{6}} 50 x^{9} \cos (5 x^{10} + 6) - e^{5 x^{6}} \sin (5 x^{10} + 6)$

$e^{5 x^{6}} 50 x^{9} \cos (5 x^{10} + 6) + 30 x^{5} e^{5 x^{6}} \sin (5 x^{10} + 6)$

$e^{5 x^{6}} 50 x^{9} \cos (5 x^{10} + 6) + e^{5 x^{6}} \sin (5 x^{10} + 6)$

None of these

I don't know

13 of 22

Differentiate

$\frac{e^{4 x + 2 x^{3}}}{7 x^{2} + 8 x^{3}}$
with respect to x.

$\frac{(4 + 6 x^{2}) e^{4 x + 2 x^{3}}}{14 x + 24 x^{2}}$

$\frac{e^{4 x + 2 x^{3}} (- 14 x + 4 x^{2} + 32 x^{3} + 42 x^{4} + 48 x^{5})}{{(7 x^{2} + 8 x^{3})}^{2}}$

$\frac{e^{4 x + 2 x^{3}} (14 x + 52 x^{2} + 32 x^{3} + 42 x^{4} + 48 x^{5})}{{(7 x^{2} + 8 x^{3})}^{2}}$

$\frac{e^{4 x + 2 x^{3}} (- 7 x + 20 x^{2} + 32 x^{3} + 14 x^{4} + 16 x^{5})}{{(7 x^{2} + 8 x^{3})}^{2}}$

None of these!

I don't know!

14 of 22

The expression C is given as:

$C = \int_{-1}^{1} (x^{4} + e^{- 3 x}) d x$

Find the value of C correct to 5 decimal places.

15 of 22

Find by substitution, the value J of the integral:

$J = \int_{4}^{8} 4 x {(4 x^{2} + 6)}^{2} d x$

The value of J is
Input your answer correct to 2 decimal places, with a minus sign if need be.

16 of 22

Assuming sufficiently small values of |x|, input the value of C in the following expansion:

${(1 + 3 x)}^{2 / 5} = \dots + \frac{216}{C} x^{3} + \dots$

17 of 22

Find the (possibly rounded) value of

$\int_{3}^{-2} 2 x^{3} d x$

-130

-30

-32.5

65

None of these!

I don't know!

18 of 22

Evaluate

$\int_{7}^{2} 10 \cos (-8 x / 5) d x$

$16 [\sin (\frac{56}{5}) - \sin (\frac{16}{5})]$

$10 [\sin (\frac{16}{5}) - \sin (\frac{196}{5})]$

$- \frac{25}{4} [- \sin (\frac{16}{5}) + \sin (\frac{56}{5})]$

$10 [\sin (\frac{56}{5}) + \sin (\frac{16}{5})]$

None of these

I don't know

19 of 22

Darlene wants to integrate this definite integral. You have to help her to find the answer.

$\int_{6}^{2} \frac{4 x^{4}}{2 + 3 x^{5}} d x$

Give your answer correct to 4 decimal places.

The answer is

20 of 22

Using integration by parts evaluate

$\int 4 x \exp (5 x) d x$

$\frac{4 exp (5 x)}{5} (x + \frac{1}{5}) + C$

$\frac{4 exp (5 x)}{5} (x - \frac{1}{5}) + C$

$\frac{4 x^{2} \exp (5 x)}{10} + C$

$\exp (5 x) (20 x - 4) + C$

None of these

I don't know

21 of 22

Evaluate:

$\int_{2}^{6} \frac{4 x}{{(- 1 - x^{2})}^{4}} d x$

If your answer is not an integer, give it correct to 5 decimal places.

My answer is

22 of 22

Evaluate the given indefinite integral using partial fractions.

$\int \frac{- 5 x^{} + 6}{7 x^{} (13 x^{} + 1)} d x$

$\frac{1}{7} [6 x^{- 1} -83 {(13 x^{} + 1)}^{- 1}] + C$

$\frac{1}{7} [6 ln x + \frac{83}{13} ln (13 x^{} + 1)] + C$

$\frac{1}{7} [6 ln x -83 ln (13 x^{} + 1)] + C$

$- \frac{1}{14} [6 x^{- 2} - \frac{83}{13} {(13 x^{} + 1)}^{- 2}] + C$

None of these!

I don't know!

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