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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 3.33a
Chapter 3, Problem 3.33a

Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = 8x; a = −3

Verified step by step guidance
1
Step 1: Identify the function f(x) = 8x and the point a = -3 where you need to find the derivative.
Step 2: Recall that the derivative of a linear function f(x) = mx is f'(x) = m. In this case, m = 8.
Step 3: Since the derivative of f(x) = 8x is constant, f'(x) = 8 for all x.
Step 4: Evaluate the derivative at the given point a = -3. Since f'(x) = 8 for all x, f'(-3) = 8.
Step 5: Conclude that the derivative of the function at the point a = -3 is f'(-3) = 8.

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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 defined as the limit of the average rate of change of the function as the interval approaches zero. In practical terms, the derivative at a point gives the slope of the tangent line to the function at that point.
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Derivatives

Tangent Lines

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line is equal to the derivative of the function at that point. This concept is crucial for understanding how functions behave locally around specific values.
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Slopes of Tangent Lines

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. In the context of derivatives, evaluating the function at a point helps in calculating the derivative at that point. For example, in the function f(x) = 8x, evaluating at a = -3 allows us to find the slope of the tangent line at that specific x-value.
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Evaluating Composed Functions
Related Practice
Textbook Question

Derivatives and tangent lines

a. For the following functions and values of a, find f′(a).

f(x) = √2x+1; a= 4

Textbook Question

Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>

a. (f^-1)'(4)

Textbook Question

Use the definition of the derivative to determine d/dx(ax²+bx+c), where a, b, and c are constants.

Textbook Question

Derivatives using tables Let h(x)=f(g(x))h(x)=f(g(x)) and p(x)=g(f(x))p(x)=g(f(x)). Use the table to compute the following derivatives.

<IMAGE>

a. h′(3)h^{\(\prime\)}\(\left\)(3\(\right\))

Textbook Question

City urbanization City planners model the size of their city using the function A(t) = - 1/50t² + 2t +20, for 0 ≤ t ≤ 50, where A is measured in square miles and t is the number of years after 2010.

a. Compute A'(t). What units are associated with this derivative and what does the derivative measure?

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Textbook Question

Find an equation of the line tangent to the following curves at the given value of x.

y = csc x; x = π/4