Derivate Chapter Test
Chapter
Chapter 2
Section
Derivate Chapter Test
Solutions 18 Videos

Which of the following is not a derivative rule? Justify your answer with an example.

A \displaystyle \frac{d}{d x}[f(x)+g(x)]=\frac{d}{d x}[f(x)]+\frac{d}{d x}[g(x)]

B \displaystyle \frac{d}{d x} f[g(x)]=\frac{d}{d x}[f(x)] \frac{d}{d x}[g(x)]

\displaystyle \begin{aligned} \text { C } \frac{d}{d x}\left[\frac{f(x)}{g(x)}\right]=& f(x) \frac{d}{d x}\left([g(x)]^{-1}\right) \\ &+[g(x)]^{-1} \frac{d}{d x}[f(x)] \end{aligned}

D \displaystyle \frac{d}{d x}[c f(x)]=c \frac{d}{d x}[f(x)]

Q1

Which statement is always true for an object moving along a vertical straight line? Explain why each of the other statements is not true.

A. The object is speeding up when v(t)a(t) is negative.

B. The object is slowing down when v(t)a(t) is positive.

C. The object is moving upward when v(t) is positive.

D. The object is at test when the acceleration is zero.

Q2

Which of the following are incorrect derivatives for \displaystyle y=\frac{-4 x}{x^{2}+1}  ? Justify your answers.

A \displaystyle \quad y^{\prime}=\frac{-4}{2 x}

B \displaystyle y^{\prime}=\frac{\left(x^{2}+1\right)(-4)-4 x(2 x)}{\left(x^{2}+1\right)^{2}}

C \displaystyle y^{\prime}=-4\left(x^{2}+1\right)^{-1}+8 x^{2}\left(x^{2}+1\right)^{-2}

D \displaystyle y^{\prime}=\frac{\left(x^{2}+1\right)(-4)+4 x(2 x)}{\left(x^{2}+1\right)^{2}}

Q3

Determine f''(3) for the function f(x) = (5x^2 -3x)^2.

Q4

Differentiate and simplify.

\displaystyle y = (3x^6)^{\frac{1}{3}} 

Q5a

Differentiate and simplify.

\displaystyle y = (x^2 -4)(2x + 1) 

Q5b

Differentiate and simplify.

\displaystyle y = -5x^3 + \frac{4}{x^5} + 1.7\pi 

Q6a

Differentiate and simplify.

\displaystyle y = (8x^2 -3x)^3 

Q6b

Differentiate and simplify.

\displaystyle y = \sqrt{9-2x}(x^2 + \frac{2}{x^3}) 

Q6c

Differentiate and simplify.

\displaystyle y = \frac{3x -2}{\sqrt{1 - x^2}} 

Q6d

Mia shoots an arrow upward with an initial vertical velocity of 11 m/s from a platform that is 2 m high. The height, h, in metres, of the arrow after t seconds can be modelled by the function h(t) = -4.9t^2 +11t + 2, t \geq 0

a) Determine the velocity and acceleration of the arrow at t = 3 s.

b) When is the arrow moving upward? When is it moving downward? Justify your answer.

c) When is the arrow momentarily at test?

d) What is the height of the arrow for the time found in part c)? What is the significance of this value?

e) When does the arrow hit the ground? With what velocity does it hit the ground?

Q7

Determine an equation for the tangent to the curve \displaystyle y = \frac{-x}{(3x + 2)^3}  at the point where x= -1.

Q8

Determine the coordinates of the point on the graph of f(x) = \sqrt{2x + 1} where the tangent line is perpendicular to the line 3x + y + 4 = 0.

Q9

The graph below shows the position function of a vehicle.

a) Is the vehicle going faster at A or at E? Is it going faster at C or at H?

b) What is the velocity of the vehicle at B and at D?

c) What happens between F and G?

d) Is the vehicle speeding up or slowing down at C and I?

e) What happens at J?

f) State whether the acceleration is positive, negative, or zero over each interval.

• i. 0 to A
• ii. B to C
• iii. D to E
• iv. F to G
• v. I to J
Q10

Student council normally sells 1500 school T—shirts for \$12 each. This year they plan to decrease the price of the T—shirts. Based on student feedback, they know that for every \$0.50 decrease in price, 20 more T—shirts will be sold.

a) Determine the demand, or price, function.

b) Determine the marginal revenue from the sales of 1800 T—shirts.

c) The cost, C, in dollars, of producing x T—shirts can be modelled by the function C(x) = -0.0005x^2 + 7.5x + 200. Determine the marginal cost of producing 1800 T-shirts.

d) Determine the actual cost of producing th e1801 st T-shirt.

e) Determine the profit and marginal profit from the sale of 1800 T—shirts.

Q11

Suppose the function \displaystyle V(t) = \frac{100 000 +5t}{1 + 0.02t}  represents the value, in dollars, of a new motorboat t years after it is purchased.

a) What is the rate of change of the value of the motorboat at 1, 3, and 6 years?

b) What was the initial value of the motorboat?

Q12

The cost, C, in dollars, of manufacturing x MP3 players per day can be modelled by the function C(x) = 0.01x^2 + 42x + 300, 0 \leq x \leq 300.
The demand function is p(x)=130 -0.4x.

a) Determine the marginal cost at a production level of 250 players.

b) Determine the actual cost of producing the 251st player.

c) Compare and describe your results from parts a) and b).

d) Determine the revenue function and the profit function.

e) Determine the marginal revenue and marginal profit for the sale of 250 players.

f) Interpret the values in part e) for this situation.

Q13

The value, V, in dollars, of an antique solid wood dining set t years after it is purchased can be modelled by the function

\displaystyle V(t) = \frac{500 +6t^3}{\sqrt{0.002t^2 + 1}}  where \displaystyle t \geq 0 .

a) What was the purchase price of the dining set.

b) Determine the rate of change of the value of the dining set after 2 years.

c) Is the value of the dining set increasing or decreasing? justify your answer.

d) What is the dinning set worth at 3 years at 10 years?

e) Compare V'(3) and V'(10). Interpret these values for this situation.