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}  3x 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}  3x is given by the first derivative y '.
y ' = 3x^{ 2}  3

We now find all values of x for which y ' = 0.
3x^{ 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}  3x
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 = x^{3}  3x below with the tangent lines.
Problem 2: Find a and b so that the line y = 3x + 4 is tangent to the graph of y = ax^{3} + bx 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 = ax^{3} + bx and y = 3x + 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 = ax^{3} + bx at x = 1
y ' = 3ax^{2} + x = 3 at x = 1.

The above gives a second equation in a and b
3a + b = 3

Solve the system of equations a + b = 1 and 3a + b = 3 to find a and b
a = 2 and b = 3.

See graphs of y = ax^{3} + bx, with a = 2 and b = 3, and y = 3x + 4 below.
Problem 3: Find conditions on a and b so that the graph of y = ae^{ 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 = ae^{ x} + bx. Find y '
y ' = ae^{ 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)
ae^{ 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 = ae^{ x} + bx has NO horizontal tangent line if a/b <= 0
Exercises
1  Find all points on the graph of y = x^{ 3}  3x where the tangent line is parallel to the line whose equation is given by y = 9x + 4.
2  Find a and b so that the line y = 2 is tangent to the graph of y = ax^{2} + bx at x = 1.
3  Find conditions on a, b and c so that the graph of y = ax^{ 3} + bx^{ 2} + cx 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  4b^{ 2}  12 ac = 0
More references on
calculus problems  
Step by Step Math Worksheets SolversNew !
Linear ProgrammingNew !
Online Step by Step Calculus Calculators and SolversNew !
Factor Quadratic Expressions  Step by Step CalculatorNew !
Step by Step Calculator to Find Domain of a Function New !
Free Trigonometry Questions with Answers

Interactive HTML5 Math Web Apps for Mobile LearningNew !

Free Online Graph Plotter for All Devices
Home Page 
HTML5 Math Applets for Mobile Learning 
Math Formulas for Mobile Learning 
Algebra Questions  Math Worksheets

Free Compass Math tests Practice
Free Practice for SAT, ACT Math tests

GRE practice

GMAT practice
Precalculus Tutorials 
Precalculus Questions and Problems

Precalculus Applets 
Equations, Systems and Inequalities

Online Calculators 
Graphing 
Trigonometry 
Trigonometry Worsheets

Geometry Tutorials 
Geometry Calculators 
Geometry Worksheets

Calculus Tutorials 
Calculus Questions 
Calculus Worksheets

Applied Math 
Antennas 
Math Software 
Elementary Statistics
High School Math 
Middle School Math 
Primary Math
Math Videos From Analyzemath
Author 
email
Updated: February 2015
Copyright © 2003  2016  All rights reserved