# Calculus 1 : How to find slope by graphing functions

## Example Questions

### Example Question #11 : How To Find Slope By Graphing Functions

What is the slope of a line tangent to  at ?

It is impossible to be tangent to that line at that point.

Explanation:

A line is tangent to a point on a curve when its slope is equivalent to the slope of the curve at the point of intersection. Therefore to solve this equation, the slope of the curve at  must be found.

The slope of a graph at any point can be found by taking the first derivative.  To take the derivative of this equation, we must use the power rule,

.

We also must remember that the derivative of an constant is 0.  By taking the first derivative of the graph equation, we obtain the slope equation

.

Plugging in , we find the slope at that point is 13, therefore any line tangent to the curve at that point must have a slope of 13.

### Example Question #12 : How To Find Slope By Graphing Functions

What is the slope of the curve  at the point  ?

Explanation:

To find the a general formula for the slope of the function , derive the function with respect to :

The slope of the function at  is then found as:

### Example Question #13 : How To Find Slope By Graphing Functions

What is the slope of the line tangent to the function at ?

None of these

Explanation:

The slope of a tangent line to a function at a point can be found by taking the derivative of the function and plugging in the point at which the slope is to be found. The derivative of the funtion can be found using the product rule.

The derivative of  is .

Also, the derivative of a sin function is the cos function.

### Example Question #14 : How To Find Slope By Graphing Functions

Find the slope of the line tangent to the function at .

None of these

Explanation:

The slope of the tangent at a certain x is the value of the derivative at that point. The derivative of  is .

The derivative of the given function is

.

Plugging in x=3 gives

.

### Example Question #15 : How To Find Slope By Graphing Functions

What is the slope of the tangent line to at

Explanation:

The derivative of a function describes the slope. Therefore, you must first find the derivative of the function, which is .

Then, you plug in the specific x value given in the problem, which is -1:

.

### 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.

.

AND