# GMAT Math : Tangent Lines

## Example Questions

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### Example Question #1 : Tangent Lines

Determine the equation of the line tangent to the curve    at the point    ?

Explanation:

To find the equation of a line tangent to a curve at a certain point, we first need to find the slope of the curve at that point. To find the slope of a function at any point, we need its derivative:

Now we can plug in the x value of the given point to find the slope of the function, and therefore the slope of tangent line, at that point:

Now that we have our slope, we can simply plug this value in with the given point to solve for the y intercept of the tangent line:

We have calculated the slope of the tangent line and its y intercept, so the equation for the line tangent to the curve    at the point    in standard form is:

### Example Question #1 : Calculating The Equation Of A Tangent Line

Determine the equation of the line tangent to the following curve at the point  .

Explanation:

First we find the slope of the tangent line by taking the derivative of the function and plugging in the -value of the point where we want to know the slope:

Now that we know the slope of the tangent line, we can plug it into the equation for a line along with the coordinates of the given point in order to calculate the -intercept:

We now have  and , so we can write the equation of the tangent line:

### Example Question #3 : Tangent Lines

Find the equation of a line tangent to the curve  at the point .

None of the above equations

Explanation:

To find the equation of a line tangent to the curve  at the point , we must first find the slope of the curve at the point  by solving the derivative  at that point:

Given the slope, we can now plug the given point and the slope at that point into the slope-intercept form of the tangent line  and solve for the -intercept :

Given our slope, the chosen point, and the -intercept, we have the equation of our tangent line:

### Example Question #4 : Tangent Lines

Find the equation of a line tangent to the curve  at the point .

None of the above

Explanation:

To find the equation of a line tangent to the curve  at the point , we must first find the slope of the curve at the point  by solving the derivative  at that point:

Given the slope, we can now plug the given point and the slope at that point into the slope-intercept form of the tangent line  and solve for the -intercept :

Given our slope, the chosen point, and the -intercept, we have the equation of our tangent line:

### Example Question #5 : Tangent Lines

Determine the equation of the tangent line to the following curve at the point   :

Explanation:

First find the slope of the tangent line by taking the derivative of the function and plugging in the x value of the given point to find the slope of the curve at that location:

So the slope of the tangent line to the curve at the given point is  .  The next step is to plug this slope into the formula for a line, along with the coordinates of the given point, to solve for the value of the y intercept of the tangent line:

We now know the slope and y intercept of the tangent line, so we can write its equation as follows:

### Example Question #6 : Tangent Lines

Find the equation of a line tangent to the curve  at the point .

None of the above

Explanation:

To find the equation of a line tangent to the curve  at the point , we must first find the slope of the curve at the point  by solving the derivative  at that point:

Given the slope, we can now plug the given point and the slope at that point into the slope-intercept form of the tangent line  and solve for the -intercept :

Given our slope, the chosen point, and the -intercept, we have the equation of our tangent line:

### Example Question #1 : Calculating The Slope Of A Tangent Line

Find the slope of the line tangent to  at .

Explanation:

f(x) in this case gives us a parabola. If we factor the f(x) we get:

This equation has one zero and that is at x=3. This also means we have a minimum at x=3.

The slope of a line tangent to a minimum or maximum is always 0, so our slope is zero.

### Example Question #2 : Calculating The Slope Of A Tangent Line

Calculate the slope of the tangent line to the following curve at the point  .

Explanation:

The slope of the tangent line to a curve at any point is simply the slope of the curve at that point. To find the slope of the function at any point, we take the derivative:

Now we can plug in the x value of the given point, , which gives us the slope of the tangent line to the curve at that point:

### Example Question #3 : Calculating The Slope Of A Tangent Line

Calculate the slope of the line tangent to the curve  at the point  .

Explanation:

To calculate the slope of a tangent line at a particular point, we need to know the slope of the curve at that point. In order to find the slope of a curve at any point, we need to calculate its derivative:

The derivative describes the slope of the curve at any point, so we need to plug in the  value of the given point    to find the slope of the tangent line at that location:

### Example Question #4 : Calculating The Slope Of A Tangent Line

Calculate the slope of the line tangent to the following curve at   :