**Background Information**

Can you answer the following 10 questions? You may wish to preview parallel, perpindicular or neither prior to trying this exercise.

Two non-vertical lines are parallel when and only when their slopes are equal and they have different y-intercept. Two non-vertical lines are perpendicular when and only when the product of their slopes is -1. Also, two lines are parallel if they have the same slope and are perpendicular if the slopes are negative reciprocals of each other.

**Example:**

When is the tangent line to y = ln x parallel to the line 2y - x = 3

**Solution:**
First, we find the slope of the line.

We solve for y, to try and get the line in the form y = mx + b.

2y - x = 3

2y = x + 3

y = (1/2)x + 3/2

Therefore, the line has a slope of 1/2

Then, we will need to obtain an equation for the slope of the tangent to the curve by differentiating.

ƒ(*x*)=ln*x*

ƒ' (*x*) = l/*x*

We want to know when these two values are the same, so we set them equal and
solve:

1/*x* = 1/2

2 = *x*

When x = 2, the tangent to y = ln x will be parallel to our line.

Ready to try a few?

Use calculus to solve the following word problems involving parallel and
perpendicular lines.

1.) When is the tangent line to ƒ(*x*)= *x* ^{2} parallel to y = 2*x*?

2.) When is the tangent line to ƒ(*x*)= *x* ^{2} perpindicular to y = 2*x*?

3.) When are the tangent lines of the following functions perpendicular?

ƒ(*x*)=3*x*^{2} + 2*x* + 1

*g*(*x*)=2*x*^{2} + 3*x* + 1

4.) When are the tangent lines of the following functions parallel?

ƒ(*x*)=3*x*^{2} + 2*x* + 1

*g*(*x*)=2*x*^{2} + 3*x* + 1

5.) When are the tangent lines of the following functions parallel?

ƒ(*x*)=√*x*

*g*(*x*)=x^{2}

6.) When are the tangent lines of the following functions perpendicular?

ƒ(*x*)=√*x*

*g*(*x*)=x^{2}

7.) When are the tangent lines of the following functions perpendicular?

ƒ(*x*)=ln *x*

*g*(*x*)=x^{2} + l

8.) At what value of x are the tangent lines to the following two functions parallel?

ƒ(*x*)=ln *x*

*g*(*x*)=x^{2} + l

9.) At what value of x are the tangent lines to the following two functions parallel?

ƒ(*x*)=3ln *x*

*g*(*x*)= 1*n*3*x*

10.) At what value of x are the tangent lines to the following two functions perpendicular?

ƒ(*x*)=3ln *x*

*g*(*x*)= 1*n*3*x*

Answers are all here in PDF.

You may also wish to try the parallel and perpindicular online calculators, although limited, they do come in handy when you don't have access to your graphing calculator.