Dawson College
Calculus 1 (201-103-05)
Winter 2019, Test 2
Read every question very carefully and write your solutions legibly for what I cannot read will be considered
incorrect. Show all your work step by step. Turn off your cell phone.
1) [8marks] Find the derivative of the function 𝑓 by using the rules of differentiation. Do NOT simplify the
answer.
a) 𝑓 𝑥 = 2𝑥 + 𝑥
c) 𝑓 𝑥 = 𝑒 −
1
3𝑥 2 − 𝑥
b) 𝑓 𝑥 =
3𝑥+4 5
𝑥 −2
d) 𝑓 𝑥 = ln 𝑥 2 + 5
𝑥
𝑑𝑦 𝑑𝑢
𝑑𝑦
2) [3marks] Find 𝑑𝑢 , 𝑑𝑥 and 𝑑𝑥 where 𝑢 is the inner function.
𝑦=
5
3𝑥 2 + 2𝑥
3) [3 marks] Find the slope and an equation of the tangent line to the curve 𝑦 = 𝑥 2𝑥 2 + 7 at the point
(3,15).
4) [3marks] Find the point(s) on the graph of
𝑓 𝑥 = 𝑥 3 + 2𝑥 2 − 5𝑥 + 3 where the tangent line is horizontal.
5) [9marks] Given the weekly cost function 𝐶 𝑥 = 4000 + 5𝑥 − 0.0003𝑥 2
a) What is the actual cost incurred in producing the 1001 st unit of product.
b) What is the marginal cost when 𝑥 = 1000?
c) Calculate 𝐶 ′ (2000) and interpret your answer.
d) Find the average cost function.
e) Find the marginal average cost function.
f) Given 𝑝𝑟𝑖𝑐𝑒 = −0.0006𝑥 + 12, find the revenue function.
g) Find the profit function.
h) Calculate 𝑃′ 7500 . Should weekly production level be increased?
i) What should the weekly production level be to achieve maximum profit?
6) Find y ′ using implicit differentiation if x 2 y 3 − xy = 8.
7) The demand equation for a certain company is given by:
81x 2 + 16p2 = 26820
where𝑥 is the demand in thousands of units and𝑝 in the unit price. How fast is the quantity demanded
increasing when the unit price is $6 and it is decreasing at a rate of $0.25 per unit per week?
8) [3marks] Two ships leave the same port at noon. Ship A sails north at 20 mph and ship B sails east at 25
mph. How fast is the distance between the ships changing at 2:00 pm?
9) [3marks] Find the derivative of the function 𝑓 𝑥 =
𝑥−5
𝑥2
by logarithmic differentiation.