Chapter 2: Q. 45 (page 222)

Describe the derivatives of each of the piecewise-defined functions in Exercises 45–48.
$f (x) = {2^{x}, if x \leq - 2; \frac{1}{x^{2}}, if x > - 2}$

Short Answer

Expert verified

The derivative of fucntion is

$f' (x) = \ln 2 (2^{x}), i f x < - 2 u n d e f i n e d, i f x = - 2 - 2 x^{- 3}, i f x > - 2$

Step by step solution

Step 1. Given Information

The given function is $f (x) = {2^{x}, if x \leq - 2; \frac{1}{x^{2}}, if x > - 2}$

Step 2. Calculation

From the differentiation rules,

$\frac{d}{d x} (2^{x}) = \ln 2 \cdot 2^{x} \frac{d}{d x} (x^{- 2}) = - 2 x^{- 3}$

$Let g (x) = 2^{x}, h (x) = \frac{1}{x^{2}} Check g (- 2) = h (- 2) g (- 2) = 2^{- 2} = \frac{1}{4} h (- 2) = \frac{1}{{(- 2)}^{2}} = \frac{1}{4}$

It follows that the function is continuous at $x = - 2$ .

$Check g' (- 2) = h' (- 2) g' (- 2) = \ln 2 {(2)}^{- 2} = \frac{\ln 2}{4} h' (- 2) = - 2 {(- 2)}^{- 3} = \frac{1}{4}$

Thus, although function is continuous at $x = 0$ , it is not differentiable.

Thus, derivative of function is as follows,

$f' (x) = \ln 2 (2^{x}), i f x < - 2 u n d e f i n e d, i f x = - 2 - 2 x^{- 3}, i f x > - 2$

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Describe the derivatives of each of the piecewise-defined functions in Exercises 45–48.
$f (x) = {2^{x}, if x \leq - 2; \frac{1}{x^{2}}, if x > - 2}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

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Describe the derivatives of each of the piecewise-defined functions in Exercises 45–48.f(x)={2x,ifx≤-2;1x2,ifx>-2}

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Discrete Mathematics

Statistics

Decision Maths

Pure Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Describe the derivatives of each of the piecewise-defined functions in Exercises 45–48.
$f (x) = {2^{x}, if x \leq - 2; \frac{1}{x^{2}}, if x > - 2}$