# Solve Tangent Lines Problems in Calculus

Tangent lines problems and their solutions, using first derivatives, are presented.

## Problem 1

Find all points on the graph of y = x 3 - 3 x where the tangent line is parallel to the x axis (or horizontal tangent line).
Solution to Problem 1:
• Lines that are parallel to the x axis have slope = 0. The slope of a tangent line to the graph of y = x 3 - 3 x is given by the first derivative y '.
y ' = 3 x 2 - 3
• We now find all values of x for which y ' = 0.
3 x 2 - 3 = 0
• Solve the above equation for x to obtain the solutions.
x = - 1 and x = 1
• The above values of x are the x coordinates of the points where the tangent lines are parallel to the x axis. Find the y coordinates of these points using y = x 3 - 3 x
for x = - 1 , y = 2
for x = 1 , y = - 2
• The points at which the tangent lines are parallel to the x axis are: (-1 , 2) and (1 , -2). See the graph of y = x3 - 3 x below with the tangent lines. ## Problem 2

Find a and b so that the line y = - 3 x + 4 is tangent to the graph of y = a x3 + b x at x = 1.
Solution to Problem 2:
• We need to determine two algebraic equations in order to find a and b. Since the point of tangency is on the graph of y = a x3 + b x and y = - 3 x + 4, at x = 1 we have
a(1)3 + b(1) = - 3(1) + 4
• Simplify to write an equation in a and b
a + b = 1
• The slope of the tangent line is -3 which is also equal to the first derivative y ' of y = a x3 + b x at x = 1
y ' = 3 a x2 + x = - 3 at x = 1.
• The above gives a second equation in a and b
3 a + b = -3
• Solve the system of equations a + b = 1 and 3 a + b = - 3 to find a and b
a = - 2 and b = 3.
• See graphs of y = a x3 + b x, with a = - 2 and b = 3, and y = - 3 x + 4 below. ## Problem 3

Find conditions on a and b so that the graph of y = a e x + bx has NO tangent line parallel to the x axis (horizontal tangent).
Solution to Problem 3:
• The slope of a tangent line is given by the first derivative y ' of y = a e x + bx. Find y '
y ' = a e x + b
• To find the x coordinate of a point at which the tangent line to the graph of y is horizontal, solve y ' = 0 for x (slope of a horizontal line = 0)
a e x + b = 0
• Rewrite the above equation as follows
e x = - b/a
• The above equation has solutions for -a/b >0. Hence, the graph of y = a e x + bx has NO horizontal tangent line if -a/b <= 0

## Exercises

1) Find all points on the graph of y = x
3 - 3 x where the tangent line is parallel to the line whose equation is given by y = 9 x + 4.
2) Find a and b so that the line y = - 2 is tangent to the graph of y = a x
2 + b x at x = 1.
3) Find conditions on a, b and c so that the graph of y = a x
3 + b x 2 + c x has ONE tangent line parallel to the x axis (horizontal tangent).

### Solutions to the Above Exercises

1) (2 , 2) and (-2 , -2)
2) a = 2 and b = - 4
3) 4 b
2 - 12 a c = 0
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