1.2 Rates of change
Consider the two graphs shown.
Which graph has a constant slope or rate of change?
Since one of the two graphs cannot have a rate of change, what is the step to be done to calculate its average rate of change?
What do we call the x-values within that straight line?
Which graph will have the same value of rate of change and average rate of change within [0,2]?
Using the answers from the previous parts, describe the difference of rate of change and average rate of change.
State whether the rate of change or the average rate of change is asked to find.
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y | 0 | 4 | 8 | 12 |
[-1,0]
[0,2]
| x | 3 | 6 | 8 | 13 |
|---|---|---|---|---|
| y | -12 | -15 | -17 | -22 |
[3,6]
[6,13]
[1,2]
[2,4]
[-3,-1]
[-4,0]
[0,3]
[3,10]
State the expression of the average rate of change, \dfrac{\Delta x}{\Delta y}, within the given interval of each graph.
[-2, 2]
[-1, 1]
[0, 2]
[-3,2]
Calculate the average rate of change of the following functions within their given interval.
f(x) = 4^x - 5,\, [-2,4]
h(x) = 3x^2 + 2x - 1,\, [1,4]
d(t) = 4t^3 - 2t^2 + 7, \, [-1, 2]
h(y) = \sqrt{y + 3},\, [0,5]
p(x)=\dfrac{1}{x^2}, \,[2,4]
r(x)=\dfrac{x^2-4}{x-2}, \, [1,3]
Hint: simplify the function first
For each of the following graphs, find the instantaneous rate of change.
Calculate the estimated instantaneous rate of change of the line tangent to the following functions.
f \left( x \right) = - \dfrac{2}{x - 3} at x = 1
f \left( x \right) = \dfrac{5}{x} at x = 2
\\ f \left( x \right) = - \dfrac{\left(x - 8\right)^{2}}{3} + 7 at x=5
\\ f \left( x \right) = - \dfrac{\left(x - 4\right)^{2}}{2} + 6 at x = 6
\\ f \left( x \right) = x^{3} - 3 x^{2} + 2 x - 1 at x = 2
g(x) = x^2 - 6x + 8 at x = 3
The function p(z) = \dfrac{1}{z+1} represents the population of a certain species over time. Find the average rate of change of the population over the interval [1, 3].
Consider the function q(r) = r^3 - 2r^2 + r - 1. What is the average rate of change of this function over the interval [-1, 1]?
Consider the function y = 2^{x}:
Find the average rate of change over the interval [0,1].
Find the average rate of change over the interval [1,2].
Find the average rate of change over the interval [2,3].
Hence, complete the following table:
| \text{Interval} | [3,4] | [4,5] | [5,6] |
|---|---|---|---|
| \text{Average rate of change} |
Explain the differences of the average rate of change and instantaneous rate of change in terms of the following.
Size of interval
Form of line
Explain why we can only approximate the instantaneous rate of change of a line tangent to a function given only the function and a point.
For the function f(x) = x^3 - 4x^2 + 2x + 1, estimate the instantaneous rate of change at the point x = 2 using the average rate of change over the intervals following intervals.
[1.9, 2.1]
[1.99, 2.01]
[1.999, 2.001]
The value of a stock is given by the function p(t) = 3t^2 + 2t + 1, where p is the price in dollars and t is the time in days. Find the average rate of change from day 3 to day 6.
State the interval given in the statement.
State the expression of the average rate of change within the interval from part (a).
The graph below represents the function f(x). Identify which of the following intervals has the greatest rate of change and which interval has the smallest rate of change.
Intervals:
[-6,-2]
[0,2]
[2,4]
[4,7]
Two functions g(x) and h(x) are given below:
g(x) = x^2 - 4x + 5
h(x) = -x^3 + 3x^2 - 2x + 1
Calculate the average rate of change of each function over the interval [1,3].
Which function has a greater rate of change over this interval?
The table below shows the number of people attending a concert at different times:
| Time (hours) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| People | 50 | 110 | 230 | 410 | 550 |
Calculate the rate of change of people attending the concert between each consecutive hour.
A car is traveling along a straight road. The table below shows the distance traveled by the car at different times:
| Time (seconds) | 0 | 5 | 10 | 15 | 20 |
|---|---|---|---|---|---|
| Distance (meters) | 0 | 25 | 60 | 105 | 160 |
Calculate the average rate of change of the distance traveled by the car for each 5-second interval.
A car is driving on a straight road at a constant speed. The table below shows the distance the car has traveled at various times. Calculate the average rate of change of the distance with respect to time between the following intervals:
| t \text{(seconds)} | 0 | 5 | 10 | 15 | 20 |
|---|---|---|---|---|---|
| d \text{(meters)} | 0 | 25 | 50 | 75 | 100 |
Between t = 0 and t = 5 seconds
Between t = 10 and t = 20 seconds
The temperature of a cup of tea is recorded every minute. The table below shows the temperature of the tea at different times. Estimate the instantaneous rate of change of the temperature with respect to time at t = 4 minutes using average rates of change over small intervals.
| t \text{(minutes)} | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| T \text{(degrees Celsius)} | 90 | 85 | 79 | 73 | 68 | 63 |
The graph below shows the height, h(t), in meters of a hot air balloon above the ground as a function of time t, in minutes.
Compare the rates of change of the hot air balloon's height at t = 2 minutes and t = 6 minutes.
A company's profit, P, in thousands of dollars, is modeled by the function P(t) = 3t^2 - 5t + 10, where t is the number of years since the company was founded.
Estimate the instantaneous rate of change of the company's profit at t = 3 years using average rates of change over small intervals.
Interpret the meaning of this rate of change in the context of the company's profit.
Consider the function f(x) = x^2 - 6x + 8.
Determine the intervals where the rate of change is positive.
Determine the intervals where the rate of change is negative.
The temperature, T(x), in a city is given by the function T(x) = 2x^3 - 15x^2 + 36x - 20, where x is the number of hours since midnight.
Determine the intervals during which the temperature is increasing.
Determine the intervals during which the temperature is decreasing.