Derivatives Chapter Review
Chapter
Chapter 2
Section
Derivatives Chapter Review
Solutions 28 Videos

Differentiate each function. State the derivative rules you used.

\displaystyle f(t) = t^3 -2t^2 + \frac{1}{t^2} 

Q1a

Differentiate each function. State the derivative rules you used.

\displaystyle f(n) = -n^5 + 5n^3 + \sqrt[3]{n^2} 

Q1b

Differentiate each function. State the derivative rules you used.

\displaystyle f(r) = r^6 - \frac{2}{5\sqrt{r}}+ r - 1 

Q1c

Air is being pumped into a spherical balloon. The volume, V, in cubic centimetres, of the balloon is V = \frac{4}{3}\pi r^3, where the radius, r, is in centimetres.

a) Determine the instantaneous rate of change of the volume of the balloon when its radius is 1.5 cm, 6 cm, and 9 cm.

b) Sketch a graph of the curve and the tangents corresponding to each radius in part a).

c) Find equations for the tangent lines.

Q2

Differentiate using the product rule.

\displaystyle f(x) = (5x +3)(2x - 11) 

Q3a

Differentiate using the product rule.

\displaystyle f(t) = (2t^2 + \sqrt[3]{t})(4t -5) 

Q3b

Differentiate using the product rule.

\displaystyle f(x) = (-1.5x^6 + 1)(3 - 8x) 

Q3c

Differentiate using the product rule.

\displaystyle f(n) = (11n + 2)(-5 + 3n^2) 

Q3d

Determine an equation for the tangent to the graph of each curve at the point that corresponds to the given value of x.

\displaystyle y = (6x -3)(-x^2 + 2), x = 1 

Q4a

Determine an equation for the tangent to the graph of each curve at the point that corresponds to the given value of x.

\displaystyle y = (-3x + 8)(x^3 -7), x =2 

Q4b

Determine \displaystyle f''(-2)  for \displaystyle f(x) = (4-x^2)(3x + 1) .

Q5

A toy missile is shot into the air. Its height, 17, in metres, after t seconds can be modelled by the function h(t) = -4.9t^2 + 15t + 0.4, \displaystyle t\geq 0 .

a) Determine the height of the toy missile at 25.

b) Determine the rate of change of the height of the toy missile at 1 s and at 4s.

c) How long does it take the toy missile to return to the ground?

d) How fast was the toy missile travelling when it hit the ground? Explain your reasoning.

Q6

The population, p, of a certain type of berry bush in a conservation area can be modelled by the function p(t) =\sqrt[3]{16t + 50t^3}, where t is time, in years.

a) Determine the rate of change of the number of berry bushes at t = 5 years.

b) When will there be 40 berry bushes?

c) What is the rate of change of the berry bush population at the time found in part b)?

Q7

Find \frac{dy}{dx} using Leibniz notation.

\displaystyle y = u^2 + 3u, u = \sqrt{x - 1}, x =5 

Q8a

Find \frac{dy}{dx} using Leibniz notation.

\displaystyle y = \sqrt{2u}, u =6-x, x =-3 

Q8b

Find \frac{dy}{dx} using Leibniz notation.

\displaystyle y = 8u(1- u), u = \frac{1}{x}, x =4 

Q8c

Determine the slope of the tangent to each function at the indicated value.

\displaystyle y = \frac{2x^2}{x + 1}  at x =2.

Q9a

Determine the slope of the tangent to each function at the indicated value.

\displaystyle y = \frac{\sqrt{3x}}{x^2 - 4}  at x =3.

Q9b

Determine the slope of the tangent to each function at the indicated value.

\displaystyle y = \frac{5x +3}{x^3 + 1}  at x =-2.

Q9c

Determine the slope of the tangent to each function at the indicated value.

\displaystyle y = \frac{-4x+ 2}{3x^2 -7x -1}  at x =1.

Q9d

Differentiate.

\displaystyle q(x) = \frac{-7x + 2}{(4x^2 -3)^3} 

Q10a

Differentiate.

\displaystyle q(x) = \frac{8x^3}{\sqrt{3x -2}} 

Q10b

Differentiate.

\displaystyle q(x) = \frac{-x + 2)^2}{(3 + 5x)^4} 

2.10mins
Q10c

Differentiate.

\displaystyle q(x) = \frac{(x^2 -3)^2}{\sqrt{4x + 5}} 

Q10d

Differentiate.

\displaystyle q(x) = \frac{(2\sqrt{x} + 7)^3}{(x^3 -3x^2 + 1)^7} 

Q10e

Determine an equation for the tangent to the curve \displaystyle y= (\frac{x^2 -1}{4x + 7})^3  at the point where x = -2.

Q11

A music store sells an average of 120 music CDs per week at $24 each. A market survey indicates that for each$0.75 decrease in price, five additional CDs will be sold per week.

a) Determine the demand, or price, function.

b) Determine the marginal revenue from the weekly sales of 150 music CDs.

c) sic CDs can be modelled by the function \displaystyle C(x) = -0.003x^2 + 4.2 x + 3000 . Determine the marginal cost of producing 150 CDs.

d) Determine the marginal profit from the weekly sales of 150 music CDs.

Q12

The voltage, V, in volts, across a resistor in an electrical circuit is V=IR, where I=4.85-0.01t^2 is the current through the resistor, in amperes; R = 15.0 + 0.11 t is the resistance, in ohms; and t is time, in seconds.

a) Write an equation for V in terms of t.

b) Determine V'(t) and interpret it for this situation.

c) Determine the rate of change of voltage at 2 s.

d) What is the rate of change of current at 2 s?

e) What is the rate of change of resistance at 2s?

f) Is the product of the values in parts d) and e) equal to the value in part b)?