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AP Calculus BC : Instantaneous Rate of Change, Average Rate of Change, and Linear Approximation

Study concepts, example questions & explanations for AP Calculus BC

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Example Questions

Example Question #1 : Instantaneous Rate Of Change, Average Rate Of Change, And Linear Approximation

Calculate the derivative of f(x)=4x^2-x+2 at the point x=1 .

Possible Answers:

Correct answer:

Explanation:

There are 2 steps to solving this problem.

First, take the derivative of f(x) .

Then, replace the value of x with the given point.

For example, if x=a , then we are looking for the value of f'(x=a) , or the derivative of f(x) at x=a .

f(x)=4x^2-x+2

Calculate f'(x)

Derivative rules that will be needed here:

Derivative of a constant is 0. For example, $\frac{\mathrm{d} }{\mathrm{d} t} 5 = 0$
Taking a derivative on a term, or using the power rule, can be done by doing the following: $\frac{\mathrm{d} }{\mathrm{d} t} t^n = n * t^{n-1}$

f'(x)=8x-1

Then, plug in the value of x and evaluate

f'(x=1)=8*1-1 = 7

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Example Question #2 : Instantaneous Rate Of Change, Average Rate Of Change, And Linear Approximation

Evaluate the first derivative if

f(x)=2e^x and x=0 .

Possible Answers:

f'(0)=0

f'(0)=e

f'(0)=1

f'(0)=2

Correct answer:

f'(0)=2

Explanation:

First we must find the first derivative of the function.

Because the derivative of the exponential function is the exponential function itelf, or

$\frac{\mathrm{d} }{\mathrm{d} x}[e^x]=e^x$

and taking the derivative is a linear operation,

we have that

$f'(x)=\frac{\mathrm{d} }{\mathrm{d} x}[f(x)]$

$=\frac{\mathrm{d} }{\mathrm{d} x}[2e^x]$

$=2\frac{\mathrm{d} }{\mathrm{d} x}[e^x]$

=2e^x

Now setting x=0

f'(0)=2e^0=2*1=2

Thus

f'(0)=2

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Example Question #1 : Instantaneous Rate Of Change, Average Rate Of Change, And Linear Approximation

Find the rate of change of f(x) when x=3.

f(x)=3e^x-x^3+12x^2

Possible Answers:

72.41

-27.2

105.26

123.23

Correct answer:

105.26

Explanation:

Find the rate of change of f(x) when x=3.

f(x)=3e^x-x^3+12x^2

To find a rate of change, we need to find the derivative.

First, recall the following rules:

$1) \frac{d}{dx}e^x=e^x$

$2)\frac{d}{dx} x^n =nx^{n-1}$

We can apply these two derivative rules to our function to get our first derivative. Then we need to plug in 3 for x and solve.

f(x)=3e^x-x^3+12x^2

f'(x)=3e^x-3x^2+24x

f'(3)=3e^3-3(3)^2+24(3)=105.26

So, our answer is 105.26

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AP Calculus BC : Instantaneous Rate of Change, Average Rate of Change, and Linear Approximation

Study concepts, example questions & explanations for AP Calculus BC

All AP Calculus BC Resources

Example Questions

Example Question #1 : Instantaneous Rate Of Change, Average Rate Of Change, And Linear Approximation

Example Question #2 : Instantaneous Rate Of Change, Average Rate Of Change, And Linear Approximation

Example Question #1 : Instantaneous Rate Of Change, Average Rate Of Change, And Linear Approximation

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