Math III · F-IF.6

Calculating and Interpreting Average Rate of Change for Advanced Function Types

Average rate of change measures how fast one quantity changes relative to another, even when the function is curved, restricted, or non-linear.

Concept Functions

Domain Interpreting Functions

Read time 7 minutes

What this learning objective is really asking you to learn

This objective asks students to calculate, estimate, and interpret average rate of change for advanced function types. Students already learned that average rate of change is

\[change in output / change in input\].

In function notation, from $x=a$ to $x=b$, it is

\[[f(b) - f(a)]/(b - a)\].

For a line, the average rate of change is the same on every interval. For advanced functions, the average rate usually depends on the interval. A square-root function may increase quickly at first and then slow. An exponential function may grow slowly at first and then rapidly. A rational function may change dramatically near an asymptote. A polynomial may increase on one interval and decrease on another.

This objective extends the same rate idea across these function families. Students must be able to compute from formulas, estimate from graphs, read from tables, and interpret in context with units.

The graph interpretation is the slope of a secant line connecting two points on the graph. Even if the graph curves between the points, the average rate of change summarizes the net change across the interval.

For example, if $f(x)=x^2$, the average rate of change from $x=1$ to $x=4$ is

\[(16 - 1)/(4 - 1) = 15/3 = 5\].

From $x=4$ to $x=7$, it is

\[(49 - 16)/(7 - 4) = 33/3 = 11\].

The rate changes because the function is not linear.

This objective is about measuring change honestly in nonlinear situations.

Why students should learn this math

Students should learn average rate of change because the world is full of changing quantities, and many do not change linearly. A phone battery may drain quickly under heavy use and slowly under light use. A population may grow exponentially. A medication concentration may decay nonlinearly. Average cost may drop quickly at first and then level off. A square-root model may increase but slow down. A rational function may behave wildly near a restriction.

If students only understand constant rates, they will misread curved behavior. Average rate of change lets them compare intervals even when the rate is not constant. It answers questions like: How much did the output change per unit input over this time span? Was the change faster in the first interval or the second? Is the model speeding up or slowing down? What is the average growth per year?

This is also the bridge to calculus. Calculus studies instantaneous rate of change, the slope at a point. Average rate of change is the slope across an interval. As the interval shrinks, the average rate approaches the instantaneous rate when the function is smooth. Students do not need limits yet, but they are building the foundation.

In practical contexts, average rates are often what people can measure. A car's average speed over a trip, a company's average monthly growth, a plant's average growth per week, or a medication's average decrease per hour can all be computed from two measurements. Even when the underlying process varies, the average rate gives useful summary information.

The “why” is that rate of change is the mathematics of motion, growth, decline, cost, speed, and trend. Math III extends that measurement to more realistic curved models.

The historical machinery: from secants to derivatives

Average rate of change is historically connected to slope and eventually calculus. The slope of a line was understood geometrically. For curves, mathematicians studied secant lines, which cut through a curve at two points, and tangent lines, which touch at one point.

The average rate of change over an interval is the slope of a secant line. Calculus developed the idea of instantaneous rate by letting the two points move closer together. Newton and Leibniz used these ideas to study motion, velocity, acceleration, and changing quantities.

Even before calculus, average rates were essential in travel, astronomy, trade, and measurement. People needed to compare distance per time, cost per item, yield per acre, and growth per year. Advanced functions simply make the rate behavior more complex.

The historical lesson is that average rate of change is not a minor graph feature. It is the doorway into the mathematics of change.

Where this fits in the big map of mathematics

This objective extends average rate of change from Math I and Math II into Math III function families. Objective 024 introduced the concept. Objective 155 applies it to rational, radical, polynomial, exponential, logarithmic, and other functions.

It connects to graph interpretation. Average rate is secant slope.

It connects to domain. The interval must lie inside the function's domain.

It connects to modeling. Units matter: dollars per item, meters per second, bacteria per hour, percent per year, cost per unit.

It connects to calculus. Average rates lead to instantaneous rates.

It connects to data analysis. Tables often provide values at intervals, and average rate summarizes change between them.

The big-map role is change measurement. Students learn to quantify how nonlinear functions change over intervals.

How to execute the skill technically

Use the formula:

\[average rate of change = [f(b) - f(a)]/(b - a)\].

Steps:

Identify interval $[a,b]$.
Compute or estimate $f(a)$ and $f(b)$.
Subtract outputs.
Subtract inputs.
Divide.
Include units.
Interpret in context.

Example with square root:

\[f(x)=\sqrt{x}\].

From $x=4$ to $x=9$:

$f(4)=2$, $f(9)=3$.

Average rate:

\[(3 - 2)/(9 - 4) = 1/5\].

So the output increases by an average of 0.2 units per 1 input unit over that interval.

From $x=9$ to $x=16$:

\[(4 - 3)/(16 - 9) = 1/7\].

The rate is smaller, showing the square-root function is increasing more slowly.

Example with exponential:

\[g(t)=100(1.1)^t\].

From $t=0$ to $t=5$:

\[g(0)=100\].

\[g(5)=100(1.1)^5 ≈ 161.05\].

Average rate:

\[(161.05 - 100)/5 ≈ 12.21\].

The quantity increased by an average of about 12.21 units per time period over the first five periods.

Worked example: average cost model

Average cost is

\[A(x)=1000/x+20\].

Find the average rate of change from $x=50$ to $x=100$.

\[A(50)=1000/50+20=40\].

\[A(100)=1000/100+20=30\].

Average rate:

\[(30 - 40)/(100 - 50) = -10/50 = -0.2\].

Interpretation: from 50 to 100 units, average cost decreases by an average of $0.20 per additional unit produced.

This does not mean every extra unit lowers average cost by exactly 20 cents. The rate varies across the interval. It is an average decrease.

Worked example from a graph

Suppose a graph shows a medicine concentration of 80 mg/L at hour 1 and 30 mg/L at hour 6. The average rate of change is

\[(30 - 80)/(6 - 1) = -50/5 = -10\].

Interpretation: from hour 1 to hour 6, concentration decreased by an average of 10 mg/L per hour.

The negative sign is meaningful. It indicates decrease. The units are also essential: mg/L per hour.

Comparing intervals

For nonlinear functions, average rate depends on interval. That is why students should always state the interval. Saying “the average rate is 5” is incomplete. From where to where? With what units? For which function?

A complete sentence: “From day 2 to day 8, the plant height increased by an average of 1.5 centimeters per day.” This sentence gives interval, quantity, direction, magnitude, and units.

Average rate of change for logarithmic and rational functions

For a logarithmic example, let

\[f(x)=log_{2}(x)\].

From $x=1$ to $x=8$,

$f(1)=0$ and $f(8)=3$.

Average rate of change:

\[(3-0)/(8-1)=3/7\].

From $x=8$ to $x=64$,

$f(8)=3$ and $f(64)=6$.

Average rate:

\[(6-3)/(64-8)=3/56\].

The function is increasing in both intervals, but the average rate is much smaller over the later interval. This captures the slow-growth nature of logarithms.

For a rational example, let

\[A(x)=500/x+10\].

From $x=50$ to $x=100$, $A$ changes from 20 to 15, so average rate is

\[(15-20)/(100-50)=-5/50=-0.1\].

From $x=100$ to $x=200$, $A$ changes from 15 to 12.5, so average rate is

\[(12.5-15)/(200-100)=-2.5/100=-0.025\].

The average cost is still decreasing, but more slowly. This is exactly the type of interpretation students need.

Secant line interpretation

The average rate of change is the slope of the secant line through two graph points. A secant line is not the same as the curve. It summarizes the curve over an interval. This matters because students often look at a curved graph and ask for “the slope.” A curve does not have one slope over its whole domain. It has average slopes over intervals and, later in calculus, instantaneous slopes at points.

The app should show the secant line dynamically. Let students drag endpoints on a graph and watch the slope update. For nonlinear functions, the changing secant slope makes the concept visible.

Unit interpretation

If $f(t)$ is measured in meters and $t$ is measured in seconds, average rate is meters per second. If $C(x)$ is dollars and $x$ is items, average rate is dollars per item. If $P(t)$ is people and $t$ is years, average rate is people per year.

Students should never submit a bare number. A correct answer includes direction, interval, units, and context. For example: “From 100 to 200 units, average cost decreased by $0.025 per additional unit.” That is far better than “-0.025.”

Problem Library

Problems in the App From This Objective

189 problems across 15 archetypes in the app.

evaluate endpoints and divide differences.

12 problems Warmup Practice Mixed Review Assessment

Problem 1

Calculate average rate of change for polynomial f(x)=x^3 on interval [1,3].

Problem 2

Calculate average rate of change for polynomial f(x)=x^2-4x on interval [0,5].

Problem 3

Calculate average rate of change for polynomial f(x)=2x^4 on interval [-1,1].

Problem 4

Calculate average rate of change for polynomial f(x)=-x^3+6x on interval [1,4].

Problem 5

Calculate average rate of change for polynomial f(x)=x^2+3x on interval [0,2].

Problem 6

Calculate average rate of change for polynomial f(x)=x^3-x on interval [-2,0].

Open in simulator

Problem 7

Calculate average rate of change for polynomial f(x)=4x^2-1 on interval [1,3].

Problem 8

Calculate average rate of change for polynomial f(x)=-2x^2+5x on interval [0,4].

Problem 9

Calculate average rate of change for polynomial f(x)=x^4+2x on interval [-1,2].

Problem 10

Calculate average rate of change for polynomial f(x)=3x-7 on interval [2,5].

Problem 11

Calculate average rate of change for polynomial f(x)=x^2 on interval [-3,3].

Problem 12

Calculate average rate of change for polynomial f(x)=x^3-2x^2+x on interval [0,1].

evaluate valid endpoints and avoid restrictions.

15 problems Warmup Practice Mixed Review Assessment

Problem 13

Calculate average rate of change for rational function f(x)=1/x on interval [1,4].

Problem 14

Calculate average rate of change for rational function f(x)=8+500/x on interval [10,20].

Problem 15

Calculate average rate of change for rational function f(x)=(x+1)/(x-2) on interval [3,5].

Open in simulator

Problem 16

Calculate average rate of change for rational function f(x)=1/(x-1) on interval [0,2].

Problem 17

Calculate average rate of change for rational function f(x)=1/x on interval [2,5].

Problem 18

Calculate average rate of change for rational function f(x)=(x+3)/x on interval [1,3].

Problem 19

Calculate average rate of change for rational function f(x)=1/(x+1) on interval [0,3].

Problem 20

Calculate average rate of change for rational function f(x)=1/(x+1) on interval [-2,0].

Problem 21

Calculate average rate of change for rational function f(x)=x^2/(x-1) on interval [2,3].

Problem 22

Calculate average rate of change for rational function f(x)=x^2/(x-1) on interval [0,2].

Problem 23

Calculate average rate of change for rational function f(x)=10-20/x on interval [2,5].

Problem 24

Calculate average rate of change for rational function f(x)=1/(2-x) on interval [0,1].

Problem 25

Calculate average rate of change for rational function f(x)=1/(2-x) on interval [1,3].

Problem 26

Calculate average rate of change for rational function f(x)=(2x+1)/(x+3) on interval [0,2].

Problem 27

Calculate average rate of change for rational function f(x)=(2x+1)/(x-1) on interval [0,2].

evaluate endpoints in domain.

12 problems Warmup Practice Mixed Review Assessment

Problem 28

Calculate average rate of change for radical function f(x)=sqrt(x) on interval [1,9].

Problem 29

Calculate average rate of change for radical function f(x)=sqrt(x-4) on interval [4,8].

Problem 30

Calculate average rate of change for radical function f(x)=cuberoot(x) on interval [-8,8].

Problem 31

Calculate average rate of change for radical function f(x)=sqrt(10-x) on interval [1,10].

Problem 32

Calculate average rate of change for radical function f(x)=sqrt(x) on interval [4,25].

Problem 33

Calculate average rate of change for radical function f(x)=sqrt(x+5) on interval [-1,4].

Open in simulator

Problem 34

Calculate average rate of change for radical function f(x)=sqrt(25-x) on interval [0,25].

Problem 35

Calculate average rate of change for radical function f(x)=cuberoot(x-1) on interval [2,28].

Problem 36

Calculate average rate of change for radical function f(x)=sqrt(4x) on interval [1,9].

Problem 37

Calculate average rate of change for radical function f(x)=sqrt(2x+1) on interval [0,4].

Problem 38

Calculate average rate of change for radical function f(x)=cuberoot(8x) on interval [-1,8].

Problem 39

Calculate average rate of change for radical function f(x)=sqrt(x) on interval [9,49].

evaluate and interpret nonlinear change.

15 problems Warmup Practice Mixed Review Assessment

Problem 40

Calculate average rate of change for exponential or logarithmic function f(x)=2^x on interval [1,4].

Problem 41

Calculate average rate of change for exponential or logarithmic function f(t)=100(1.1)^t on interval [0,2].

Problem 42

Calculate average rate of change for exponential or logarithmic function f(x)=ln(x) on interval [1,e^2].

Open in simulator

Problem 43

Calculate average rate of change for exponential or logarithmic function f(x)=log_2(x) on interval [2,8].

Problem 44

Calculate average rate of change for exponential or logarithmic function f(x)=3^x on interval [0,2].

Problem 45

Calculate average rate of change for exponential or logarithmic function f(x)=5(2)^x on interval [1,3].

Problem 46

Calculate average rate of change for exponential or logarithmic function f(x)=100(0.5)^x on interval [0,2].

Problem 47

Calculate average rate of change for exponential or logarithmic function f(x)=ln(x) on interval [e,e^3].

Problem 48

Calculate average rate of change for exponential or logarithmic function f(x)=log(x) on interval [10,100].

Problem 49

Calculate average rate of change for exponential or logarithmic function f(x)=log_3(x) on interval [3,27].

Problem 50

Calculate average rate of change for exponential or logarithmic function f(x)=e^x on interval [0,1].

Problem 51

Calculate average rate of change for exponential or logarithmic function f(x)=2ln(x) on interval [1,e^2].

Problem 52

Calculate average rate of change for exponential or logarithmic function f(x)=4^(-x) on interval [0,1].

Problem 53

Calculate average rate of change for exponential or logarithmic function f(x)=ln(x+1) on interval [0,e-1].

Problem 54

Calculate average rate of change for exponential or logarithmic function f(x)=3^x+5 on interval [0,1].

read endpoint coordinates and compute.

12 problems Warmup Practice Mixed Review Assessment

Problem 55

Estimate average rate of change from graph endpoints (2,5) and (6,17).

Problem 56

Estimate average rate of change from graph endpoints (1,10) and (4,4).

Problem 57

Estimate average rate of change from graph endpoints (-3,8) and (3,8).

Problem 58

Estimate average rate of change from graph endpoints (0,1.5) and (5,9).

Problem 59

Estimate average rate of change from graph endpoints (1,1) and (5,9).

Open in simulator

Problem 60

Estimate average rate of change from graph endpoints (-2,7) and (2,-1).

Problem 61

Estimate average rate of change from graph endpoints (0,0) and (10,5).

Problem 62

Estimate average rate of change from graph endpoints (-4,-3) and (0,5).

Problem 63

Estimate average rate of change from graph endpoints (3,12) and (7,0).

Problem 64

Estimate average rate of change from graph endpoints (-1,-5) and (4,5).

Problem 65

Estimate average rate of change from graph endpoints (0,10) and (2,4).

Problem 66

Estimate average rate of change from graph endpoints (-5,2) and (5,2).

select endpoints and compute change quotient.

12 problems Warmup Practice Mixed Review Assessment

Problem 67

Estimate average rate of change from table values f(1)=4, f(3)=10, f(7)=18 over interval [3,7].

Problem 68

Estimate average rate of change from table values f(0)=100, f(2)=80, f(5)=20 over interval [0,5].

Problem 69

Estimate average rate of change from table values f(-1)=6, f(4)=16 over interval [-1,4].

Open in simulator

Problem 70

Estimate average rate of change from table values uneven table values f(2)=9 and f(9)=30 over interval [2,9].

Problem 71

Estimate average rate of change from table values f(1)=5, f(4)=17, f(8)=25 over interval [1,4].

Problem 72

Estimate average rate of change from table values f(0)=0, f(5)=25, f(10)=100 over interval [0,10].

Problem 73

Estimate average rate of change from table values g(-2)=10, g(0)=6, g(3)=0 over interval [-2,3].

Problem 74

Estimate average rate of change from table values h(1)=100, h(3)=90, h(7)=70 over interval [3,7].

Problem 75

Estimate average rate of change from table values y(0)=10, y(2)=14, y(6)=22 over interval [0,6].

Problem 76

Estimate average rate of change from table values k(-5)=20, k(0)=10, k(5)=0 over interval [-5,5].

Problem 77

Estimate average rate of change from table values p(1)=1, p(2)=4, p(3)=9 over interval [1,3].

Problem 78

Estimate average rate of change from table values q(10)=50, q(15)=40, q(20)=30 over interval [10,20].

attach units and describe change over interval.

12 problems Warmup Practice Mixed Review Assessment

Problem 79

Interpret average rate of change -2.5 dollars per item for context average cost from 10 to 20 items over interval [10,20].

Problem 80

Interpret average rate of change 12 meters per second for context height or distance over time over interval [1,4] seconds.

Problem 81

Interpret average rate of change 0.8 pH units per hour for context chemical measurement over interval [0,5] hours.

Problem 82

Interpret average rate of change -15 people per year for context population model over interval [2,6] years.

Problem 83

Interpret average rate of change 3 degrees Celsius per hour for context temperature of a liquid over interval [0, 4] hours.

Problem 84

Interpret average rate of change -5 miles per hour per minute for context speed of a car over interval [10, 15] minutes.

Problem 85

Interpret average rate of change 0.2 feet per day for context water level in a reservoir over interval [1, 7] days.

Open in simulator

Problem 86

Interpret average rate of change 500 dollars per month for context company's monthly revenue over interval [3, 6] months.

Problem 87

Interpret average rate of change 100 bacteria per hour for context number of bacteria in a culture over interval [0, 2] hours.

Problem 88

Interpret average rate of change -0.1 gallons per mile for context fuel remaining in a tank over interval [50, 100] miles.

Problem 89

Interpret average rate of change 0.75 dollars per day for context stock price over interval [10, 20] days.

Problem 90

Interpret average rate of change -0.05 atmospheres per kilometer for context atmospheric pressure with altitude over interval [0, 5] kilometers.

compute and compare interval behavior.

12 problems Warmup Practice Mixed Review Assessment

Problem 91

Compare average rates of change over intervals [0,2] and [2,4] for function f(x)=x^2.

Problem 92

Compare average rates of change over intervals [0,1] and [4,9] for function f(x)=sqrt(x).

Problem 93

Compare average rates of change over intervals [0,1] and [3,4] for function f(x)=2^x.

Open in simulator

Problem 94

Compare average rates of change over intervals [1,2] and [2,4] for function f(x)=1/x.

Problem 95

Compare average rates of change over intervals [0,1] and [1,2] for function f(x)=x^3.

Problem 96

Compare average rates of change over intervals [0,1] and [1,3] for function f(x)=-x^2.

Problem 97

Compare average rates of change over intervals [1,e] and [e,e^2] for function f(x)=ln(x).

Problem 98

Compare average rates of change over intervals [0,2] and [5,7] for function f(x)=3x+5.

Problem 99

Compare average rates of change over intervals [0,pi/2] and [pi/2,pi] for function f(x)=sin(x).

Problem 100

Compare average rates of change over intervals [0,1] and [1,2] for function f(x)=e^x.

Problem 101

Compare average rates of change over intervals [-2,0] and [0,2] for function f(x)=x^2+3x.

Problem 102

Compare average rates of change over intervals [1,2] and [2,3] for function f(x)=1/x^2.

compare secant slopes.

12 problems Warmup Practice Mixed Review Assessment

Problem 103

Identify the interval with greatest or least average rate from rates on A,B,C are 2,-1,5.

Open in simulator

Problem 104

Identify the interval with greatest or least average rate from table f(0)=0, f(2)=4, f(5)=10, f(6)=9.

Problem 105

Identify the interval with greatest or least average rate from graph secants show steepest upward segment from x=4 to x=7.

Problem 106

Identify the interval with greatest or least average rate from all rates negative: -3,-1,-5.

Problem 107

Identify the interval with greatest or least average rate from rates for intervals P, Q, R are -4, 3, 1.

Problem 108

Identify the interval with greatest or least average rate from rates for intervals X, Y, Z are 7, -2, -6.

Problem 109

Identify the interval with greatest or least average rate from table g(1)=10, g(3)=16, g(4)=18, g(7)=12.

Problem 110

Identify the interval with greatest or least average rate from table h(0)=5, h(2)=1, h(5)=7, h(6)=4.

Problem 111

Identify the interval with greatest or least average rate from graph secants show steepest downward segment from x=1 to x=3.

Problem 112

Identify the interval with greatest or least average rate from rates are 1.5, 0.5, 2.5.

Problem 113

Identify the interval with greatest or least average rate from rates are -0.5, -2.5, -1.5.

Problem 114

Identify the interval with greatest or least average rate from table k(0)=0, k(1)=5, k(3)=5, k(5)=10.

connect calculation to graph geometry.

12 problems Warmup Practice Mixed Review Assessment

Problem 115

Relate average rate of change to the secant line through (1,2) and (5,10).

Problem 116

Relate average rate of change to the secant line through (0,7) and (3,1).

Problem 117

Relate average rate of change to the secant line through (2,4) and (6,4).

Open in simulator

Problem 118

Relate average rate of change to the secant line through (1,3) and (4,12).

Problem 119

Relate average rate of change to the secant line through (0,0) and (4,8).

Problem 120

Relate average rate of change to the secant line through (1,5) and (3,1).

Problem 121

Relate average rate of change to the secant line through (-2,3) and (5,3).

Problem 122

Relate average rate of change to the secant line through (0,1) and (4,3).

Problem 123

Relate average rate of change to the secant line through (2,7) and (5,4).

Problem 124

Relate average rate of change to the secant line through (0,0) and (2,10).

Problem 125

Relate average rate of change to the secant line through (0,10) and (2,0).

Problem 126

Relate average rate of change to the secant line through (1,1) and (3,5).

check domain and discontinuities.

15 problems Warmup Practice Mixed Review Assessment

Problem 127

Determine whether average rate is meaningful for f(x)=1/(x-2) over interval [1,3].

Problem 128

Determine whether average rate is meaningful for f(x)=sqrt(x-4) over interval [4,9].

Problem 129

Determine whether average rate is meaningful for f(x)=(x^2-1)/(x-1) over interval [0,2].

Problem 130

Determine whether average rate is meaningful for f(x)=ln(x) over interval [-1,2].

Problem 131

Determine whether average rate is meaningful for f(x)=x^3-2x+1 over interval [-5,5].

Problem 132

Determine whether average rate is meaningful for f(x)=1/(x+5) over interval [0,4].

Problem 133

Determine whether average rate is meaningful for f(x)=1/x over interval [0,5].

Open in simulator

Problem 134

Determine whether average rate is meaningful for f(x)=sqrt(x+1) over interval [-3,-2].

Problem 135

Determine whether average rate is meaningful for f(x)=sqrt(x) over interval [1,9].

Problem 136

Determine whether average rate is meaningful for f(x)=ln(x+1) over interval [0,5].

Problem 137

Determine whether average rate is meaningful for f(x)=ln(x) over interval [0,e].

Problem 138

Determine whether average rate is meaningful for f(x)=tan(x) over interval [0,pi].

Problem 139

Determine whether average rate is meaningful for f(x)=(x^2-4)/(x-2) over interval [0,1].

Problem 140

Determine whether average rate is meaningful for f(x)=(x^2-9)/(x-3) over interval [1,3].

Problem 141

Determine whether average rate is meaningful for f(x)=cbrt(x) over interval [-8,8].

solve change-quotient equation.

12 problems Warmup Practice Mixed Review Assessment

Problem 142

Find the missing endpoint value given average rate information average rate 4 on [2,7], f(2)=3.

Problem 143

Find the missing endpoint value given average rate information average rate -2 on [1,5], f(5)=6.

Problem 144

Find the missing endpoint value given average rate information average rate 0.5 on [0,10], f(0)=8.

Problem 145

Find the missing endpoint value given average rate information average rate r on [a,b], f(a)=A.

Problem 146

Find the missing endpoint value given average rate information average rate 3 on [1,6], f(1)=5.

Problem 147

Find the missing endpoint value given average rate information average rate 5 on [2,4], f(4)=18.

Problem 148

Find the missing endpoint value given average rate information average rate -1 on [0,8], f(0)=10.

Problem 149

Find the missing endpoint value given average rate information average rate -3 on [3,7], f(7)=5.

Problem 150

Find the missing endpoint value given average rate information average rate 1.5 on [2,6], f(2)=7.

Problem 151

Find the missing endpoint value given average rate information average rate 2.5 on [1,3], f(3)=12.

Problem 152

Find the missing endpoint value given average rate information average rate 10 on [5,15], f(5)=20.

Open in simulator

Problem 153

Find the missing endpoint value given average rate information average rate -5 on [10,20], f(20)=50.

compare nonlinear behavior across intervals.

12 problems Warmup Practice Mixed Review Assessment

Problem 154

Explain how average rate supports model comparison for two cost models over 10 to 20 units.

Problem 155

Explain how average rate supports model comparison for two growth models over early and late intervals.

Open in simulator

Problem 156

Explain how average rate supports model comparison for radical versus linear distance model.

Problem 157

Explain how average rate supports model comparison for rational saturation models.

Problem 158

Explain how average rate supports model comparison for exponential decay versus linear decay models.

Problem 159

Explain how average rate supports model comparison for two projectile trajectories over their flight.

Problem 160

Explain how average rate supports model comparison for logarithmic versus square root growth models.

Problem 161

Explain how average rate supports model comparison for logistic versus exponential population growth models.

Problem 162

Explain how average rate supports model comparison for a cubic polynomial versus an exponential function.

Problem 163

Explain how average rate supports model comparison for two drug concentration models over absorption and elimination phases.

Problem 164

Explain how average rate supports model comparison for algorithms with O(n log n) and O(n^2) complexity.

Problem 165

Explain how average rate supports model comparison for Newton's Law of Cooling versus a constant rate cooling model.

explain interval-based change.

12 problems Warmup Practice Mixed Review Assessment

Problem 166

Distinguish average rate from instantaneous rate in context car distance from t=2 to t=5.

Problem 167

Distinguish average rate from instantaneous rate in context curved profit graph.

Open in simulator

Problem 168

Distinguish average rate from instantaneous rate in context rational graph near asymptote.

Problem 169

Distinguish average rate from instantaneous rate in context population model over a decade.

Problem 170

Distinguish average rate from instantaneous rate in context temperature change from 8 AM to 12 PM.

Problem 171

Distinguish average rate from instantaneous rate in context stock price movement over a trading day.

Problem 172

Distinguish average rate from instantaneous rate in context water level in a tank over an hour with variable inflow.

Problem 173

Distinguish average rate from instantaneous rate in context bacterial population growth over 6 hours.

Problem 174

Distinguish average rate from instantaneous rate in context height of a ball thrown upwards from t=1 to t=3 seconds.

Problem 175

Distinguish average rate from instantaneous rate in context total cost of production for increasing units.

Problem 176

Distinguish average rate from instantaneous rate in context car's fuel consumption over a long trip.

Problem 177

Distinguish average rate from instantaneous rate in context concentration of a reactant in a chemical reaction over time.

catch endpoint, domain, discontinuity, subtraction, and unit mistakes.

12 problems Warmup Practice Mixed Review Assessment

Problem 178

Correct the average-rate error in Used f(3)-f(1) divided by 3 instead of 2 for interval [1,3].

Problem 179

Correct the average-rate error in Computed across x=0 for f(x)=1/x.

Problem 180

Correct the average-rate error in Subtracted inputs over outputs.

Problem 181

Correct the average-rate error in Reported slope with no units in a context problem.

Problem 182

Correct the average-rate error in Calculated (f(1)-f(5))/(5-1) for the interval [1,5].

Problem 183

Correct the average-rate error in Used (3-1)/(f(3)-f(1)) for the average rate of change.

Problem 184

Correct the average-rate error in Found the derivative f'(x) at x=2 instead of the average rate over [1,3].

Problem 185

Correct the average-rate error in Used 5 as the denominator for the average rate of change over [1,5].

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Problem 186

Correct the average-rate error in Computed average rate of change for ln(x) over [-2, 2].

Problem 187

Correct the average-rate error in Simplified ( (x+h)^2 - x^2 ) to (x^2 + h^2 - x^2) in the numerator.

Problem 188

Correct the average-rate error in Calculated (f(b)+f(a))/(b-a) for the average rate of change.

Problem 189

Correct the average-rate error in Used f(x) values at x=1 and x=2 for the average rate over [0,3].