1.3 Rates of change in linear and quadratic functions
Define the average rate of change for a function over the interval [a, b].
What does it mean for a function to have a constant average rate of change over any length of input-value interval?
What type of function has an average rate of change over any length of input-value interval that is constant?
What type of function has an average rate of change over consecutive equal-length input-value intervals that can be given by a linear function?
What does it mean for a function's graph to be concave up? What about concave down?
Does the function that passes through the following points: \left\{\left( - 2 , - 5 \right), \left(1, - 20 \right), \left(2, - 25 \right), \left(7, - 50 \right), \left(9, - 60 \right)\right\} have a constant or a variable rate of change?
Compare and contrast the average rate of change for the following functions on the interval from x=1 to x=5:
f(x) = 2x + 3
g(x) = x^2 - 4x + 5
For each function:
Identify whether the function is linear or quadratic.
Describe the general behavior of the average rate of change.
f(x) = 3x + 2
g(x) = -x^2 + 4x + 3
Calculate the average rate of change for the following functions over the given intervals:
f(x) = 2x - 3, interval[1, 4]
g(x) = -x + 5, interval[-2, 2]
g(x) = -2x^2 + 5x - 1, interval [-1, 2]
k(x) = \dfrac{1}{2}x^2 + x - 3, interval [-2, 3]
What is the slope of the secant line from the point (3, 9) to (5, 25) and what does it represent?
Given the linear function f(x) = 3x - 2, calculate the average rate of change of the function between the following intervals and explain how it is changing:
Interval [1, 3]
Interval [3, 5]
Given the quadratic function f(x) = x^2 - 4x + 7, calculate the average rate of change of the function between the following intervals and explain how it is changing:
Interval [1, 2]
Interval [2, 3]
Which function has an average rate of change over equal-length input-value intervals that is decreasing for all small-length intervals?f(x) = x^3 or f(x) = x^2?
Consider the function f(x) = x^3 - 3x^2 + 2x + 1.
Calculate the average rate of change of f(x) over the following intervals:
[0, 1]
[1, 2]
[2, 3]
[3, 4]
Determine the intervals where the function is concave up and concave down.
Determine if the graph of the function is concave up or concave down and explain your reasoning.
Compare the average rate of change of the function f(x) = x^2 over the intervals [1, 2] and [2, 3]. What do you notice?
Given the quadratic function g(x) = x^2 - 4x + 5:
Find the average rate of change of the function over the interval [-2, 3].
Determine the slope of the secant line for the same interval.
Graphically represent the average rate of change for the function over a given .
f(x) = x^2, interval [1, 3]
k(x) = 4x - 2, interval[0, 3]
A car rental company charges a flat fee of \$20 per day for renting a car, plus an additional \$0.15 per mile driven.
Determine the average rate of change of the cost with respect to the number of miles driven.
Discuss the slope of the secant line between two points on the cost function for different number of miles driven.
Discuss the concavity of the cost function graph. What does it tell about the rate of change of the cost with respect to miles driven?
Given the function f(x) = x^3 - x^2 + x:
Determine the intervals over which the function is increasing and decreasing.
Find the intervals over which the function is concave up and concave down.
A ball is thrown upward with an initial velocity of 60 feet per second. The height of the ball (in feet) above the ground after t seconds can be modeled by the quadratic function h(t) = -16t^2 + 60t.
Calculate the average rate of change of the height with respect to time for the interval 0 \leq t \leq 2.
Discuss the slope of the secant line between two points on the height function for different time intervals.
Discuss the concavity of the height function graph. What does it tell about the acceleration of the ball?
A linear function f(x) and a quadratic function g(x) are shown on the graph.
Estimate the average rate of change for the linear function on the interval from x=-3 to x=1.
Estimate the average rate of change for the quadratic function on the interval from x=-3 to x=1.
How do the rates of change differ for these functions?
The position of a moving object is given by the function s(t) = -2t^2 + 8t + 3, where t is the time in seconds and s(t) is the position in meters.
Calculate the average rate of change of the position between t = 1 and t = 3.
Explain the meaning of the average rate of change in this context.
Determine the slope of the secant line between t = 1 and t = 3, and explain its relationship to the average rate of change.
Consider the function \\ f \left( x \right) = - \dfrac{\left(x - 8\right)^{2}}{3} + 7 and a tangent line at x=5:
Calculate the average rate of change between x = 5 and x = 8
Calculate the instantaneous rate of change at x = 5.
Due to adverse market conditions, Tobias and Gwen have had to reduce the number of staff at their respective companies, which currently have 310 staff.
Tobias plans on reducing staff numbers by 41 each year for the next five years, while Gwen plans on reducing staff numbers by 8 next year, 16 in the year after next, 24 in the year after that, and so on.
| \text{Year} | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| \text{Tobia's} \\ \text{ staff} | ||||||
| \text{Gwen's} \\ \text{ staff} |
Complete the table. Note that year 0 represents the current year.
What will be the rate of change in the size of Tobias's staff between years 3 and 4?
What will be the rate of change in the size of Gwen's staff between years 3 and 4?
Is the function representing the size of Tobias's staff linear or non-linear? Explain your answer.
Is the function representing the size of Gwen's staff linear or non-linear? Explain your answer.
Whose staff will decrease more quickly between years 3 and 4?
Whose staff will decrease more quickly between years 10 and 11?
The graph shows the height of a cricket ball in feet after it is thrown.
Find the rate of change of the height of the ball in the interval between:
When it is thrown and t = 1.
t = 1 and when it is at its highest point.
When it is at its highest point and t = 3.
t = 3 and when it returns to the ground.
In which of the above intervals is the ball travelling at its fastest speed?
What do the negative rates of change in the interval between t = 2 and t = 3 and in the interval between t = 3 and t = 4 indicate?