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AP Calculus AB
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Textbook
Introduction
1. Limits
2. Derivative basics
2.1 The derivative
2.1.1 Limit definition of the derivative
2.1.2 Additional problem types
2.2 Tangent lines & slopes
2.3 Power rule
2.4 Product & quotient rules
2.5 Special derivatives
2.6 Differentiability & continuity
3. Advanced differentiation
4. Contextual uses
5. Analytical uses
6. Integration
7. Differential equations
8. Applications of integrals
Wrapping up
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2.1.2 Additional problem types
Achievable AP Calculus AB
2. Derivative basics
2.1. The derivative
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Additional problem types

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What you’ll learn

  • Finding rates of change from a table
  • Recognizing limit expressions as derivative definitions

Using tables

Problems that involve tabular data often appear on the AP exam.

A bathtub is being filled with water and the depth of the water is measured over a few minutes. Based on the table below:

Time t (minutes) Depth (inches)
2 3
4 8
6 14
7 16
9 19

a) Find the average rate of change over the time interval 4≤t≤7.

b) Estimate the instantaneous rate of change at t=3.

Solutions

a) Average rate of change over [4,7]

(spoiler)

In AP Calculus, time t is often the input/independent variable. Let water depth be the function of time defined by D(t).

The average rate of change of the depth of the water over the interval [4,7] is

7−4D(7)−D(4)​=316−8​=38​

The units are inches per minute because depth (numerator) is measured in inches and time (denominator) is measured in minutes.

So on average, the water depth increased by 38​ inches per minute between minutes 4 and 7.

b) Instantaneous rate of change at t=3

(spoiler)

The best estimate that can be made for the instantaneous rate of change uses the average rate of change over a small nearby interval.

Use the closest values around 3 minutes: t=2 and t=4. The average rate of change over that interval is

4−2D(4)−D(2)​=28−3​=25​ in/min

This is the best estimate for how quickly the water level was changing at exactly 3 minutes.

Using graphs

Shown below is the graph of function f.

Finding average rate of change
Finding average rate of change

If g is the function defined by

g(x)=f(3x)

find the average rate of change of g(x) over the interval [−1,34​].

Solution

(spoiler)

The average rate of change of g over [−1,34​] is

34​−(−1)g(34​)−g(−1)​

Since g(x)=f(3x),

g(34​)=f(3⋅34​)=f(4)=1​

and

g(−1)=f(−3)=4​

Then the average rate of change of g is

34​+11−4​​=37​−3​=−79​​

Finding f from a limit

Most derivative problems ask you to find f′(x) given a function f(x). However, some problems involve identifying the original function f(x) and the point a from a given limit.

The key is to recognize which limit definition is being used:

f′(a)=x→alim​x−af(x)−f(a)​orf′(a)=h→0lim​hf(a+h)−f(a)​

Examples

  1. Find a function f(x) and a number a such that the following limit represents f′(a).

h→0lim​h(3+h)2−9​

Solution

(spoiler)

Since the limit notation specifies h→0, match it to the 2nd definition

f′(a)=h→0lim​hf(a+h)−f(a)​

In the limit, the expression (3+h)2 matches f(a+h), which suggests that f(x)=x2 with a=3.

Checking,

f(3)=(3)2=9

which matches the second term in the numerator of the limit expression.

Therefore, the limit represents the derivative of f(x)=x2 evaluated at x=3.

  1. Find f(x) and a number a such that the limit represents f′(a).

x→πlim​x−πcos(x)+1​

Solution

(spoiler)

Since the limit notation specifies x→π, then a=π in the 1st limit definition

f′(a)=x→alim​x−af(x)−f(a)​

In the given limit, the expression cos(x) matches f(x), so f(x)=cos(x).

Checking f(a) with a=π,

f(π)=cos(π)=−1

so

f(x)−f(a)=cos(x)−(−1)=cos(x)+1

which matches the numerator.

Therefore, the limit represents the derivative of f(x)=cos(x) evaluated at x=π.

Working backwards from limit definitions

  • Two definitions to recognize: f′(a)=limx→a​x−af(x)−f(a)​ and f′(a)=limh→0​hf(a+h)−f(a)​
  • Identify which definition applies based on whether the limit is x→a or h→0
  • Extract f(x) and a by pattern-matching terms in the numerator, then verify by checking f(a)

Using tables for rates of change

  • Average rate of change over [a,b]: b−af(b)−f(a)​ using table values directly
  • Instantaneous rate of change at a point: use average rate of change over the smallest available surrounding interval as the best estimate
  • Always include units (e.g., inches per minute) when interpreting results
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Additional problem types

What you’ll learn

  • Finding rates of change from a table
  • Recognizing limit expressions as derivative definitions

Using tables

Problems that involve tabular data often appear on the AP exam.

A bathtub is being filled with water and the depth of the water is measured over a few minutes. Based on the table below:

Time t (minutes) Depth (inches)
2 3
4 8
6 14
7 16
9 19

a) Find the average rate of change over the time interval 4≤t≤7.

b) Estimate the instantaneous rate of change at t=3.

Solutions

a) Average rate of change over [4,7]

(spoiler)

In AP Calculus, time t is often the input/independent variable. Let water depth be the function of time defined by D(t).

The average rate of change of the depth of the water over the interval [4,7] is

7−4D(7)−D(4)​=316−8​=38​

The units are inches per minute because depth (numerator) is measured in inches and time (denominator) is measured in minutes.

So on average, the water depth increased by 38​ inches per minute between minutes 4 and 7.

b) Instantaneous rate of change at t=3

(spoiler)

The best estimate that can be made for the instantaneous rate of change uses the average rate of change over a small nearby interval.

Use the closest values around 3 minutes: t=2 and t=4. The average rate of change over that interval is

4−2D(4)−D(2)​=28−3​=25​ in/min

This is the best estimate for how quickly the water level was changing at exactly 3 minutes.

Using graphs

Shown below is the graph of function f.

If g is the function defined by

g(x)=f(3x)

find the average rate of change of g(x) over the interval [−1,34​].

Solution

(spoiler)

The average rate of change of g over [−1,34​] is

34​−(−1)g(34​)−g(−1)​

Since g(x)=f(3x),

g(34​)=f(3⋅34​)=f(4)=1​

and

g(−1)=f(−3)=4​

Then the average rate of change of g is

34​+11−4​​=37​−3​=−79​​

Finding f from a limit

Most derivative problems ask you to find f′(x) given a function f(x). However, some problems involve identifying the original function f(x) and the point a from a given limit.

The key is to recognize which limit definition is being used:

f′(a)=x→alim​x−af(x)−f(a)​orf′(a)=h→0lim​hf(a+h)−f(a)​

Examples

  1. Find a function f(x) and a number a such that the following limit represents f′(a).

h→0lim​h(3+h)2−9​

Solution

(spoiler)

Since the limit notation specifies h→0, match it to the 2nd definition

f′(a)=h→0lim​hf(a+h)−f(a)​

In the limit, the expression (3+h)2 matches f(a+h), which suggests that f(x)=x2 with a=3.

Checking,

f(3)=(3)2=9

which matches the second term in the numerator of the limit expression.

Therefore, the limit represents the derivative of f(x)=x2 evaluated at x=3.

  1. Find f(x) and a number a such that the limit represents f′(a).

x→πlim​x−πcos(x)+1​

Solution

(spoiler)

Since the limit notation specifies x→π, then a=π in the 1st limit definition

f′(a)=x→alim​x−af(x)−f(a)​

In the given limit, the expression cos(x) matches f(x), so f(x)=cos(x).

Checking f(a) with a=π,

f(π)=cos(π)=−1

so

f(x)−f(a)=cos(x)−(−1)=cos(x)+1

which matches the numerator.

Therefore, the limit represents the derivative of f(x)=cos(x) evaluated at x=π.

Key points

Working backwards from limit definitions

  • Two definitions to recognize: f′(a)=limx→a​x−af(x)−f(a)​ and f′(a)=limh→0​hf(a+h)−f(a)​
  • Identify which definition applies based on whether the limit is x→a or h→0
  • Extract f(x) and a by pattern-matching terms in the numerator, then verify by checking f(a)

Using tables for rates of change

  • Average rate of change over [a,b]: b−af(b)−f(a)​ using table values directly
  • Instantaneous rate of change at a point: use average rate of change over the smallest available surrounding interval as the best estimate
  • Always include units (e.g., inches per minute) when interpreting results

More from The derivative

  • Limit definition of the derivative