### All AP Calculus BC Resources

## Example Questions

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

Calculate the derivative of at the point .

**Possible Answers:**

**Correct 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,
- Taking a derivative on a term, or using the power rule, can be done by doing the following:

Then, plug in the value of x and evaluate

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

Evaluate the first derivative if

and .

**Possible Answers:**

**Correct answer:**

First we must find the first derivative of the function.

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

and taking the derivative is a linear operation,

we have that

Now setting

Thus

### Example Question #1 : Derivative At A Point

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

**Possible Answers:**

**Correct 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:

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

### Example Question #1 : Derivative Defined As Limit Of Difference Quotient

Evaluate .

**Possible Answers:**

**Correct answer:**

To find , substitute and use the chain rule:

So

and

### Example Question #1 : Derivative Defined As Limit Of Difference Quotient

What is the equation of the line tangent to the graph of the function

at the point ?

**Possible Answers:**

**Correct answer:**

The slope of the line tangent to the graph of at is

, which can be evaluated as follows:

The equation of the line with slope through is:

### Example Question #1 : Derivative At A Point

What is the equation of the line tangent to the graph of the function

at the point ?

**Possible Answers:**

**Correct answer:**

The slope of the line tangent to the graph of at is

, which can be evaluated as follows:

, the slope of the line.

The equation of the line with slope through is:

### Example Question #1 : Derivative At A Point

What is the equation of the line tangent to the graph of the function

at ?

**Possible Answers:**

**Correct answer:**

The slope of the line tangent to the graph of at is

, which can be evaluated as follows:

, the slope of the line.

The equation of the line with slope through is:

### Example Question #1 : Derivative At A Point

What is the equation of the line tangent to the graph of the function

at the point ?

**Possible Answers:**

**Correct answer:**

The slope of the line tangent to the graph of at the point is , which can be evaluated as follows:

The line with this slope through has equation:

### Example Question #1 : Derivative Defined As Limit Of Difference Quotient

What is the equation of the line tangent to the graph of the function

at the point ?

**Possible Answers:**

**Correct answer:**

The slope of the line tangent to the graph of at the point is , which can be evaluated as follows:

The line with slope 28 through has equation:

### Example Question #1 : Derivative At A Point

Given the function , find the slope of the point .

**Possible Answers:**

The slope cannot be determined.

**Correct answer:**

To find the slope at a point of a function, take the derivative of the function.

The derivative of is .

Therefore the derivative becomes,

since .

Now we substitute the given point to find the slope at that point.

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