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MATH1141: Calculus I A3-1

TRU Open Learning

Assignment 3 (8%)

Assignment 3 is graded with a total of 100 marks and it contributes 8 percent

towards your course grade.

[20 marks] 1. Find the derivative using the differentiation rules. You need

not simplify your answer.

(4 marks) a. 𝑦 = √𝑥2−3𝑥

2𝑥2+4𝑥−1

(4 marks) b. 𝑦 = (𝑥4 + 3)𝑒 −

1

𝑥2−2𝑥

(4 marks) c. 𝑠 = ln(tan 𝑥2) + tan(ln 𝑥2)

(4 marks) d. 𝑔 = √sec (

1

𝑥 )

4

(4 marks) e. 𝑓 = (𝑥4 + 3) ln ( 1

𝑥2−2𝑥 )

[8 marks] 2. If a tank holds 50,000 litres of water which drains from the

bottom of the tank in 40 minutes, then the volume of water, V,

which remains in the tank after t minutes is given by

𝑉(𝑡) = 50000(1 − 𝑡

40 )2 0 ≤ 𝑡 ≤ 40

(4 marks) a. Find the rate at which water is draining from the tank after

10 minutes and after 20 minutes.

(4 marks) b. At what time is the water flowing the fastest? At which

time is it flowing the slowest?

[10 marks] 3.

(6 marks) a. For the function 𝑓(𝑥) = 𝑥+1

𝑥−1 at which point(s) does the

tangent line have a slope of -1? Give exact answers.

A3-2 Assignment 3

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(4 marks) b. Find an equation of the tangent line(s) at the point(s) in a.

Give exact answers.

[8 marks] 4.

(4 marks) a. Suppose that 𝑓(𝑥) = 𝑔(𝑥3) + 𝑒 𝑔(3𝑥) where 𝑔′ and 𝑔′′ exist.

Find 𝑓′ and 𝑓′′ in terms of 𝑔′ and 𝑔′′.

(4 marks) b. Find 𝑓", when 𝑓(𝑥) = 𝑒 𝑥 2

+ 5𝑥.

[12 marks] 5. Differentiate using implicit differentiation:

(6 marks) a. √2𝑥2 − 3𝑦 = 𝑥3𝑦2 + 4. Do not square both sides.

(6 marks) b. 𝑥2 cos(3𝑦) − 𝑦2 sin(𝑥2) = 5

[12 marks] 6. Differentiate using the logarithmic differentiation:

(6 marks) a. 𝑦 = (2𝑥2−3)

3 5

(4𝑥2−3𝑥+1) 4 7(6𝑥+2)

1 2

(6 marks) b. 𝑦 = (ln 𝑥) 1

𝑥

[10 marks] 7. Suppose that the cost, C in dollars, of producing x units of a

certain item is 𝐶(𝑥) = 920 + 3𝑥 − 0.03𝑥2 + 0.0006𝑥3.

(2 marks) a. Find the marginal cost function

(4 marks) b. Find 𝐶′(100) and explain its meaning

(4 marks) c. Compare 𝐶′(100) with the cost of producing the 101st item.

MATH1141: Calculus I A3-3

TRU Open Learning

[12 marks] 8. Find the equations of both the tangent lines to the

hyperbola 𝑥2 − 4𝑦2 = 9 that pass through the point (−3, 3).

Note that the point (−3, 3) is not on the hyperbola.

[8 marks] 9.

(4 marks) a. Find lim 𝑥→0

sin𝑥 csc6𝑥

(4 marks) b. If 𝑔(𝜃) = 𝜃sin𝜃, find 𝑔′′( 𝜋

6 ). Give exact values.