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Basic definition-of-derivative calculations: Find the derivatives of the functions that follow, using (a) the h0definition of the derivative and (b) the zxdefinition of the derivative.

fx=x3

Short Answer

Expert verified

f'x=3x2

Step by step solution

01

Given Information

The given expression isfx=x3

02

Simplification

Using (a)

f'x=limh0fx+h-fxh

f'x=limh0x+h3-x3h

role="math" localid="1650970063541" f'x=limh0x3+h3+3x2h+3xh2-x3h

role="math" localid="1650970073601" f'x=limh0hh2+3x2+3xhh

role="math" localid="1650970082470" f'x=limh0h2+3x2+2xh

Applying limit

f(x)=2x2

Using (b)

role="math" localid="1650970131635" f'x=limzxfz-fxz-x

role="math" localid="1650969972120" f'x=limzxz3-x3z-x

role="math" localid="1650969988908" f'x=limzxz-xz2+x2+2zxz-x

f'x=limzxz2+x2+2zx

Applying limit

f'x=3x2

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Most popular questions from this chapter

Find the derivatives of the functions in Exercises 21–46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

f(x)=x2-x3+5x9x-1

Find the derivatives of the functions in Exercises 21–46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

f(x)=x-12(x2-1)3

Use the definition of the derivative to find f'for each function fin Exercises 34-59

f(x)=1x2

Suppose h(t) represents the average height, in feet, of a person who is t years old.

(a) In real-world terms, what does h(12) represent and what are its units? What does h' (12) represent, and what are its units?

(b) Is h(12) positive or negative, and why? Is h'(12) positive or negative, and why?

(c) At approximately what value of t would h(t) have a maximum, and why? At approximately what value of t would h' (t) have a maximum, and why?

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?

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