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Find the derivatives of each of the absolute value and piecewise-defined functions in Exercises 65-72.

f(x)=x

Short Answer

Expert verified

The derivative of the function isf'(x)=-1x<0DNEx=01x>0.

Step by step solution

01

Step 1. Given Information

The given function isf(x)=x.

02

Step 2. Rewrite the function

The given function can be rewritten as,

f(x)=-xx0xx>0

03

Step 3. Find the derivative

  • It is known that, the derivative of the identity function is ddx(x)=1.
  • Find the derivative for localid="1648725381209" x<0.

f'(x)=ddx(-x)=-1

  • Find the derivative for x>0.

f'(x)=ddx(x)=1

  • Since -11, the derivatives at left and right pieces are not equal at x=0. The derivative does not exists at x=0.
  • So, the derivative of the function is,f'(x)=-1x<0DNEx=01x>0.

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Most popular questions from this chapter

Every morning Linda takes a thirty-minute jog in Central Park. Suppose her distance s in feet from the oak tree on the north side of the park tminutes after she begins her jog is given by the function s(t)shown that follows at the left, and suppose she jogs on a straight path leading into the park from the oak tree.

(a) What was the average rate of change of Linda’s distance from the oak tree over the entire thirty-minute jog? What does this mean in real-world terms?

(b) On which ten-minute interval was the average rate of change of Linda’s distance from the oak tree the greatest: the first 10minutes, the second 10minutes, or the last10minutes?

(c) Use the graph of s(t)to estimate Linda’s average velocity during the 5-minute interval fromt=5tot=10. What does the sign of this average velocity tell you in real-world terms?

(d) Approximate the times at which Linda’s (instantaneous) velocity was equal to zero. What is the physical significance of these times?

(e) Approximate the time intervals during Linda’s jog that her (instantaneous) velocity was negative. What does a negative velocity mean in terms of this physical example?

A bowling ball dropped from a height of 400feet will be s(t)=400-16t2feet from the ground after tseconds. Use a sequence of average velocities to estimate the instantaneous velocities described below:

After t=2 seconds, with h=0.1,h=0.01h=-0.1andh=-0.01

A bowling ball dropped from a height of 400feet will be s(t)=400-16t2feet from the ground after tseconds. Use a sequence of average velocities to estimate the instantaneous velocities described below:

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Use the definition of the derivative to find f'for each function fin Exercises

f(x)=3x

For each function fgraphed in Exercises 65-68, determine the values of xat which ffails to be continuous and/or differentiable. At such points, determine any left or right continuity or differentiability. Sketch secant lines supporting your answers.

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