Calculus: Graphical, Numerical, Algebraic

3rd Edition

ISBN: 9780133688399

Author: Finney, Ross L.

Publisher: Pearson/Prentice Hall

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Question

Chapter 2.4, Problem 15E

To determine

To check: Whether the curve f(x)={2−2x−x2, x<02x+2, x≥0 has a tangent at x=0 or not. If it exists give the slope of tangent.

Expert Solution & Answer

Answer to Problem 15E

The curve f(x)={2−2x−x2, x<02x+2, x≥0 does not have a tangent at x=0 as the slope from left side is not equal to the slope from right side.

Explanation of Solution

Given information:

The curve is f(x)={2−2x−x2, x<02x+2, x≥0 and the point is x=0.

Calculation:

For the tangent of the curve f(x) to exist at x=a , the slope from the left side of the point x=a must be equal to the slope from the right side of the point x=a.

The function corresponding to the left side of the point x=0 is f(x)=2−2x−x2.

Check the slope of the curve from the left side of x=0.

limh→0−f(a+h)−f(a)h=limh→0−f(0+h)−f(0)h=limh→0−f(h)−f(0)h

Substitute h for x in the function.

f(x)=2−2x−x2f(h)=2−2h−h2

Substitute 0 for x in the function.

f(x)=2x+2f(0)=2(0)+2f(0)=0+2f(0)=2

Substitute 2−2h−h2 for f(h) and 2 for f(0) in the formula of slope.

limh→0−f(h)−f(0)h=limh→0−2−2h−h2−2h=limh→0−h(−2−h)h=limh→0−(−2−h)=−2

So, the slope of the curve f(x)={2−2x−x2, x<02x+2, x≥0 from the left side of the point x=0 is −2.

The function corresponding to the right side of the point x=0 is f(x)=2x+2.

Check the slope of the curve from the right side of x=0.

limh→0+f(a+h)−f(a)h=limh→0+f(0+h)−f(0)h=limh→0+f(h)−f(0)h

Substitute h for x in the function.

f(x)=2x+2f(h)=2(h)+2f(h)=2h+2

Substitute 0 for x in the function.

f(x)=2x+2f(0)=2(0)+2f(0)=0+2f(0)=2

Substitute 2h+2 for f(h) and 2 for f(0) in the formula of slope.

limh→0+f(h)−f(0)h=limh→0+2h+2−2h=limh→0+2hh=limh→0+(2)=2

Further simplify,

limh→0+f(h)−f(0)h=limh→0+(2−h)=2−0=2

So, the slope of the curve f(x)={2−2x−x2, x<02x+2, x≥0 from the right side of the point x=0 is equal to 2.

The slope of the given curve from the left side of point x=0 is not equal to the slope from the right side of point x=0.

Therefore, the curve f(x)={2−2x−x2, x<02x+2, x≥0 does not have a tangent at x=0.

Chapter 2.4, Problem 14E

Chapter 2.4, Problem 16E

Chapter 2 Solutions

Calculus: Graphical, Numerical, Algebraic

Ch. 2.1 - Prob. 1QR Ch. 2.1 - Prob. 2QR Ch. 2.1 - Prob. 3QR Ch. 2.1 - Prob. 4QR Ch. 2.1 - Prob. 5QR Ch. 2.1 - Prob. 6QR Ch. 2.1 - Prob. 7QR Ch. 2.1 - Prob. 8QR Ch. 2.1 - Prob. 9QR Ch. 2.1 - Prob. 10QR

Ch. 2.1 - Prob. 1E Ch. 2.1 - Prob. 2E Ch. 2.1 - Prob. 3E Ch. 2.1 - Prob. 4E Ch. 2.1 - Prob. 5E Ch. 2.1 - Prob. 6E Ch. 2.1 - Prob. 7E Ch. 2.1 - Prob. 8E Ch. 2.1 - Prob. 9E Ch. 2.1 - Prob. 10E Ch. 2.1 - Prob. 11E Ch. 2.1 - Prob. 12E Ch. 2.1 - Prob. 13E Ch. 2.1 - Prob. 14E Ch. 2.1 - Prob. 15E Ch. 2.1 - Prob. 16E Ch. 2.1 - Prob. 17E Ch. 2.1 - Prob. 18E Ch. 2.1 - Prob. 19E Ch. 2.1 - Prob. 20E Ch. 2.1 - Prob. 21E Ch. 2.1 - Prob. 22E Ch. 2.1 - Prob. 23E Ch. 2.1 - Prob. 24E Ch. 2.1 - Prob. 25E Ch. 2.1 - Prob. 26E Ch. 2.1 - Prob. 27E Ch. 2.1 - Prob. 28E Ch. 2.1 - Prob. 29E Ch. 2.1 - Prob. 30E Ch. 2.1 - Prob. 31E Ch. 2.1 - Prob. 32E Ch. 2.1 - Prob. 33E Ch. 2.1 - Prob. 34E Ch. 2.1 - Prob. 35E Ch. 2.1 - Prob. 36E Ch. 2.1 - Prob. 37E Ch. 2.1 - Prob. 38E Ch. 2.1 - Prob. 39E Ch. 2.1 - Prob. 40E Ch. 2.1 - Prob. 41E Ch. 2.1 - Prob. 42E Ch. 2.1 - Prob. 43E Ch. 2.1 - Prob. 44E Ch. 2.1 - Prob. 45E Ch. 2.1 - Prob. 46E Ch. 2.1 - Prob. 47E Ch. 2.1 - Prob. 48E Ch. 2.1 - Prob. 49E Ch. 2.1 - Prob. 50E Ch. 2.1 - 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To check: Whether the curve f ( x ) = { 2 − 2 x − x 2 , x &lt; 0 2 x + 2 , x ≥ 0 has a tangent at x = 0 or not. If it exists give the slope of tangent.