# Calculus 1 : Graphing Functions

## Example Questions

### Example Question #16 : How To Find Slope By Graphing Functions

At what  values is the slope of the tangent line to equal to zero?

Explanation:

First, you must find the slope equation of the tangent line to the function, which is just the derivative of the function:

.

Since that is the slope equation, you need to set that equal to 0 and factor:

, or , which yields .

### Example Question #17 : How To Find Slope By Graphing Functions

The coordinates of the following points are given as follows:  and  .

If a   is tangent to  at point  and  is tangent to  at point . If   , then which of the following statements is true?

Explanation:

If the two lines are perpendicular, then their slopes must be opposite reciprocals. Since both lines are tangent to the curve, , their slopes will be equal to the slope of the curve at the points to which they are tangent. So  has slope  and  has slope . Following this, if the lines are perpendicular, then

### Example Question #18 : How To Find Slope By Graphing Functions

Find the slope of the following function at .

Explanation:

This problem basically amounts to finding the derivative and evaluating it at the given value. We need to use a couple different techniques to find the derivative of this function but they're all fairly simple. We need the chain rule which says:

and the product rule, which is

So, using these we can calculate our derivative.  and , so:

this equals,

and when we plug in , we get

which can be written as

### Example Question #19 : How To Find Slope By Graphing Functions

If , what is the slope at ?

Explanation:

To find the slope, you must find the derivative of the function. Remember, when taking the derivative, multiply the exponent by the coefficient and then subtract 1 from the exponent, this is known as the power rule.

Therefore, the derivative is:

.

Then, to find the slope at 1, just plug 1 into the derivative.

.

### Find the slope of the line through:

AND

Explanation:

The slope (m) between two points is found with the following formula:

We can apply this formula with the points we are given:

This is one of the answer choices.

### Example Question #21 : How To Find Slope By Graphing Functions

Find the slope of the equation  at the point .

Explanation:

The slope of a function at a given point is found by first taking the derivative of the function.

Use the power rule to find this derivative, given by:

By evaluating the derivative when  you will find the slope of the function at the point .

### Example Question #22 : How To Find Slope By Graphing Functions

What is the slope of at ?

Explanation:

To find the value of the slope at , you must first find the derivative of the function since that will give us the slope. To take the derivative of a term, multiplty the exponent by the coefficient in front of the  term, and then subtract  from the exponent. Therefore, the derivative is: . Then, plug in  to get the correct slope value. .

### Example Question #23 : How To Find Slope By Graphing Functions

What is the slope of at

Explanation:

To find the slope, you must first find the derivative. To take the derivative, multiply the exponent by the leading coefficient and then subtract 1 from the exponent. Therefore, the derivative is: . Then, plug in 2 to get the specific value: .

### Example Question #24 : How To Find Slope By Graphing Functions

What is the slope of when

Explanation:

To find the slope, you must first find the derivative function. To take the derivative, multiply the exponent by the leading coefficient and then subtract 1 from the exponent. Therefore, the derivative is: . Then, plug in -1. .

### Example Question #25 : How To Find Slope By Graphing Functions

What is the slope at x=1 if ?