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MATH1141: Calculus I A1-1

TRU Open Learning

Assignment 1 (8%)

Assignment 1 is graded with a total of 100 marks and it contributes 8 percent

towards your course grade.

[10 marks] 1. Find 𝑓(𝑥+ℎ)−𝑓(𝑥)

ℎ for 𝑓(𝑥) = √4𝑥 + 2. Simplify your answer until

there are no radicals in the numerator.

[10 marks] 2. Find a formula for the function 𝑓 whose graph is shown in the

diagram below.

[8 marks] 3. Give the domain in interval notation for each function:

(4 marks) a. 𝑓(𝑥) = 2𝑥2−5

√𝑥2−𝑥−6

(4 marks) b. 𝑔(𝑡) = √5 − 𝑡 + √6 + 𝑡

[12 marks] 4. For 𝑓(𝑥) = √2𝑥 + 3 and 𝑔(𝑥) = 𝑥2 + 1, find 𝑓𝑜𝑔, 𝑔𝑜𝑓, 𝑓𝑜𝑓, and

𝑔𝑜𝑔 and give their domain in interval notation.

A1-2 Assignment 1

TRU Open Learning

[10 marks] 5. An electricity company charges its customers a base rate of $15

a month plus 8 cents per kilowatt-hour (kWh) for the first 1200

kWh and 12 cents per kWh for usage over 1200 kWh. Write the

cost, C, in dollars as a function of x, the number of kWh used for

0 ≤ 𝑥 < 1500 and graph the function

[8 marks] 6. A certain paper book sells for $25. The author receives a

royalty of 12% on the first 5000 copies sold, 15% on the next

10000 copies sold, and 20% for any additional copies sold. Write

R, the amount of royalties in dollars as a function of x, the

number of copies sold.

[12 marks] 7. For the function 𝑓(𝑥) = −4𝑥

5𝑥+2

(4 marks) a. Show that it is one-to-one.

(4 marks) b. Find the inverse, 𝑓 −1(𝑥).

(4 marks) c. Check that 𝑓(𝑓 −1(𝑥)) = 𝑥.

[10 marks] 8. Suppose that the population of a certain species is given by

𝑃(𝑡) = 1000

1+9𝑒−𝑡 where t is given in years.

(4 marks) a. Find the inverse of P.

(4 marks) b. Use the inverse in a. to find the time required for the

population to reach 800.

(2 marks) c. What happens to the population as time goes on, i.e. as 𝑡 → ∞?

[10 marks] 9. For the function 𝑓(𝑥) = 2+3 ln 𝑥

4−ln 𝑥

MATH1141: Calculus I A1-3

TRU Open Learning

(5 marks) a. Find 𝑓 −1(𝑥).

(5 marks) b. Check that 𝑓 −1(𝑓(𝑥)) = 𝑥.

[10 marks] 10. Find the exact value of each expression.

(3 marks) a. sin−1 (sin 5𝜋

4 )

(3 marks) b. cos−1 (𝑒 ln 1−

1

2 ln 2

)

(4 marks) c. sin(2sin−1 ( 3

5 ))