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

Card 0 of 6

Question

Evaluate the first derivative if

and .

Answer

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

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

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

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

Now setting

Thus

Compare your answer with the correct one above

Question

Calculate the derivative of at the point .

Answer

There are 2 steps to solving this problem.

First, take the derivative of .

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

For example, if , then we are looking for the value of , or the derivative of at .

Calculate

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: $\frac{\mathrm{d} }{\mathrm{d} t} t^n = n * t^{n-1}$

Then, plug in the value of x and evaluate

Compare your answer with the correct one above

Question

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

Answer

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

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.

So, our answer is 105.26

Compare your answer with the correct one above

Question

Evaluate the first derivative if

and .

Answer

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

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

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

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

Now setting

Thus

Compare your answer with the correct one above

Question

Calculate the derivative of at the point .

Answer

There are 2 steps to solving this problem.

First, take the derivative of .

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

For example, if , then we are looking for the value of , or the derivative of at .

Calculate

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: $\frac{\mathrm{d} }{\mathrm{d} t} t^n = n * t^{n-1}$

Then, plug in the value of x and evaluate

Compare your answer with the correct one above

Question

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

Answer

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

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.

So, our answer is 105.26

Compare your answer with the correct one above

Tap the card to reveal the answer