the drawing shows two situations in which charges are placed on the x and y axes

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:

Problem 2

Find a and b so that the line y = - 3 x + 4 is tangent to the graph of y = a x ³ + b x at x = 1.

Solution to Problem 2:

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).