# GMAT Math : Calculating the equation of a tangent line

## Example Questions

### 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 #2 : Tangent Lines

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