Mid Chapter Review Rates of Change
Chapter
Chapter 1
Section
Mid Chapter Review Rates of Change
Solutions 33 Videos

Calculate the product of each radical expression and its corresponding conjugate.

a. \displaystyle \sqrt{5} - \sqrt{2} 

b. \displaystyle 3\sqrt{5} + 2\sqrt{2} 

c. \displaystyle 9 + 2\sqrt{5} 

d. \displaystyle 3\sqrt{5} -2\sqrt{10} 

Q1

Rationalize each denominator.

\displaystyle \frac{5 +\sqrt{2}}{\sqrt{3}} 

Q2a

Rationalize each denominator.

\displaystyle \frac{2\sqrt{3}+ 4}{\sqrt{3}} 

Q2b

Rationalize each denominator.

\displaystyle \frac{5}{\sqrt{7} -4} 

Q2c

Rationalize each denominator.

\displaystyle \frac{2\sqrt{3}}{\sqrt{3} -2} 

Q2d

Rationalize each denominator.

\displaystyle \frac{5\sqrt{3}}{2\sqrt{3} + 4} 

Q2e

Rationalize each denominator.

\displaystyle \frac{3\sqrt{2}}{2\sqrt{3} - 5} 

Q2f

Rationalize each numerator.

\displaystyle \frac{\sqrt{2}}{5} 

Q3a

Rationalize each numerator.

\displaystyle \frac{\sqrt{3}}{6 +\sqrt{2}} 

Q3b

Rationalize each numerator.

\displaystyle \frac{\sqrt{7}-4}{5} 

Q3c

Rationalize each numerator.

\displaystyle \frac{2\sqrt{3} -5}{3\sqrt{2}} 

Q3d

Rationalize each numerator.

\displaystyle \frac{\sqrt{3} - \sqrt{7}}{4} 

Q3e

Rationalize each numerator.

\displaystyle \frac{2\sqrt{3} + \sqrt{7}}{5} 

Q3f

Determine the equation of the line described by the given information.

slope - \frac{2}{3}, passing through the point (0, 6).

Q4a

Determine the equation of the line described by the given information.

passing through points (2, 7) and (6, 11).

Q4b

Determine the equation of the line described by the given information.

parallel to y = 4x - 6, passing through point (2, 6)

Q4c

Determine the equation of the line described by the given information.

perpendicular to y = -5x + 3, passing through point (-1, -2).

Q4d

Find the slope of PQ, in simplified form, given P(1, -1) and Q(1 + h, f(1 + h)), where f(x) = -x^2.

Q5

Consider the function y = x^2 -2x -2.

a) Copy and complete the following tables of values. P and Q are points on the graph of f(x).

b) Use your results to approximate the slope of the tangent to the graph of f(x) at point P.

c) Calculate the slope of the secant where the x-coordinates of Q is -1 +h.

d) Use your results for part c to calculate the slope of the tangent to the graph of f(x) at point P.

Q6

Calculate the slope of the tangent to each curve at the given point or value of x.

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

Q7a

Calculate the slope of the tangent to each curve at the given point or value of x.

\displaystyle f(x) = \frac{1}{x} , x= \frac{1}{3} 

Q7b

Calculate the slope of the tangent to each curve at the given point or value of x.

\displaystyle f(x) = \frac{4}{x -2}, (6, 1) 

Q7c

Calculate the slope of the tangent to each curve at the given point or value of x.

\displaystyle f(x) = \sqrt{x + 4}, x = 5 

Q7d

The function s(t) = 6t(t + 1) describes the distance (in km) that a car has travelled after a time t(in hours), for 0 \leq t \leq 6.

a) Calculate the average velocity of the car during the following intervals.

• i. from t =2 to t =3
• ii. from t =2 to t =2.1
• from t =2 to t = 2.01

b) Use your results from above to approximate the instantaneous velocity of the car when t =2.

c) Find the average of the car from t =2 to t = 2+ h.

d) Use your results for part c to find the velocity when t = 2.

Q8

Calculate the instantaneous rate of change of f(x) with respect to x at the given value of x.

\displaystyle f(x) = 5- x^2, x = 2 

Q9a

Calculate the instantaneous rate of change of f(x) with respect to x at the given value of x.

\displaystyle f(x) = \frac{3}{x}, x = \frac{1}{2} 

Q9b

An oil tank is being drained for cleaning. After t minutes, there are V litres of oil left in the tank, where V(t) = 50(30 -t)^2, 0 \leq t\leq 30.

a. Calculate the average rate of change in the volume during the first 20 mins

b. Calculate the rate of change in volume at time t = 20.

Q10

Find the equation of the tangent at the given value of x.

\displaystyle y = x^2 + x - 3, x =4 

Q11a

Find the equation of the tangent at the given value of x.

\displaystyle y = 2x^2 -7, x = -2 

Q11b

Find the equation of the tangent at the given value of x.

\displaystyle f(x) = 3x^2 + 2x -5, x = -1 

Q11c

Find the equation of the tangent at the given value of x.

\displaystyle f(x) = 5x^2 - 8x + 3, x = 1 

Q11d

Find the equation of the tangent to the graph of the function at the given value of x.

\displaystyle f(x) = \frac{x}{x + 3}, x = -5 

\displaystyle y = \frac{2x + 5}{5x -1}, x = -1