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Chapter 2: Q. 0 (page 164)

Problem Zero: Read the section and make your own sum-

mary of the material.

Short Answer

Expert verified

The combination of position function and direction is defined as velocity function.

The slope of any tangent line at any point x=cis defined asf'(c)=limh鈫�0f(c+h)-f(c)h.

The line that passes through points a,f(a)&(b,f(b))on an interval (a,b)is known as the secant line.

The average rate of change becomes the instantaneous rate of change as鈭�x鈫�0.

Step by step solution

01

Step 1. Given information.

The topic of the given section is An Intuitive Introduction to Derivatives.

02

Step 2. Summary of section.

The combination of position function and direction is defined as velocity function.

The slope of any tangent line at any point x=cis defined as f'(c)=limh鈫�0f(c+h)-f(c)h.

The line that passes through points role="math" localid="1649811552168" a,f(a)&(b,f(b))on an interval (a,b)is known as the secant line.

The average rate of change becomes the instantaneous rate of change as role="math" localid="1649811585656" 鈭�x鈫�0.

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Most popular questions from this chapter

Use the definition of the derivative to find ffor each function fin Exercises 39-54

f(x)=x3x+1

A tomato plant given xounces of fertilizer will successfully bear T(x)pounds of tomatoes in a growing season.

(a) In real-world terms, what does T(5)represent and what are its units? What does T'(5)represent and what are its units?

(b) A study has shown that this fertilizer encourages tomato production when less than 20ounces are used, but inhibits production when more than 20ounces are used. When is T(x)positive and when is T(x)negative? When is T'(x)positive and when is T'(x)negative?

Taking the limit: We have seen that if f is a smooth function, then f'(c)鈮�f(c+h)-f(c)hThis approximation should get better as h gets closer to zero. In fact, in the next section we will define the derivative in terms of such a limit.

f'(c)=limh鈫�0f(c+h)-f(c)h.

Use the limit just defined to calculate the exact slope of the tangent line tof(x)=x2atx=4.

Velocity v(t) is the derivative of position s(t). It is also true that acceleration a(t) (the rate of change of velocity) is the derivative of velocity. If a race car鈥檚 position in miles t hours after the start of a race is given by the function s(t), what are the units of s(1.2)? What are the units and real-world interpretation of v(1.2)? What are the units and real-world interpretations of a(1.2)?

Suppose u(x)=3x2+1and f(u)=u2+3u51-u. Use the chain rule to find role="math" localid="1648356625815" ddx(f(u(x))) without first finding the formula for f(u(x)).
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