AP Calculus BC : Derivative at a Point

Study concepts, example questions & explanations for AP Calculus BC

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Example Questions

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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:

Explanation:

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:

Explanation:

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:

Explanation:

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:

Explanation:

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:

Explanation:

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:

Explanation:

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:

Explanation:

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:

Explanation:

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:

Explanation:

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:

Explanation:

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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