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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.2.30b
Chapter 3, Problem 3.2.30b

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

Verified step by step guidance
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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.
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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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Related Practice
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b. Graph the tangent and normal lines on the given graph.

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