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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 3 - Derivatives문제 3.2.30b
3장, 문제 3.2.30b

21–30. Derivatives
b. Evaluate f'(a) for the given values of a.
f(t) = 3t⁴; a= -2, 2

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1
Step 1: Identify the function f(t) = 3t^4 and the values of a for which we need to evaluate the derivative, which are a = -2 and a = 2.
Step 2: Find the derivative of the function f(t) with respect to t. Use the power rule for differentiation, which states that if f(t) = t^n, then f'(t) = n*t^(n-1).
Step 3: Apply the power rule to f(t) = 3t^4. The derivative f'(t) is found by multiplying the exponent by the coefficient and reducing the exponent by one.
Step 4: Substitute the given values of a into the derivative f'(t) to find f'(-2) and f'(2).
Step 5: Simplify the expressions obtained from substituting a = -2 and a = 2 into f'(t) to find the values of the derivative at these points.

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주요 개념

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Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the curve of the function at a given point. The derivative can be computed using various rules, such as the power rule, product rule, and quotient rule, depending on the form of the function.
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가이드 코스
05:44
Derivatives

Power Rule

The power rule is a basic differentiation rule used to find the derivative of functions of the form f(t) = t^n, where n is a real number. According to this rule, the derivative f'(t) is given by n*t^(n-1). This rule simplifies the process of differentiation, especially for polynomial functions, making it easier to evaluate derivatives at specific points.
추천 영상:
5:50
Power Rules

Evaluating Derivatives at Specific Points

Evaluating the derivative at specific points involves substituting the given values into the derivative function. For instance, once the derivative f'(t) is calculated, substituting a = -2 or a = 2 into f'(t) yields the slope of the tangent line at those points on the original function. This process is crucial for understanding the behavior of the function at specific locations.
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04:50
Critical Points
관련 실천
교과서 질문

A race Jean and Juan run a one-lap race on a circular track. Their angular positions on the track during the race are given by the functions θ(t) and ϕ(t), respectively, where 0≤t≤4 and t is measured in minutes (see figure). These angles are measured in radians, where θ=ϕ=0 represent the starting position and θ=ϕ=2π represent the finish position. The angular velocities of the runners are θ′(t) and ϕ′(t). <IMAGE>

b. Which runner has the greater average angular velocity?

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교과서 질문

Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>

b. Find equations of all lines tangent to the curve y(x²+4)=8 when y=1.

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교과서 질문

62–65. {Use of Tech} Graphing f and f'

b. Compute and graph f'.

f(x)=e^−x tan^−1 x on [0,∞)

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교과서 질문

Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.


b. ln(x + 1) + ln(x − 1) = ln(x² − 1), for all x.

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교과서 질문

Use a graphing utility to graph the curve and the tangent line on the same set of axes.

y = (x + 5) / (x - 1); a = 3

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교과서 질문

45–50. Tangent lines Carry out the following steps. <IMAGE>

b. Determine an equation of the line tangent to the curve at the given point.

(x²+y²)²=25/4 xy²; (1, 2)

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