Textbook Question
A circle has an initial radius of 50 ft when the radius begins decreasing at a rate of 2 ft/min. What is the rate of change of the area at the instant the radius is 10 ft?
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.2.2
If f′(x)=3x+2, find the slope of the line tangent to the curve y=f(x) at x=1, 2, and 3.
Verified step by step guidance1
Step 1: Understand that the derivative of a function, f′(x), represents the slope of the tangent line to the curve y=f(x) at any point x.
Step 2: Recognize that you are given f′(x) = 3x + 2, which is the expression for the slope of the tangent line at any point x on the curve.
Step 3: To find the slope of the tangent line at x = 1, substitute x = 1 into the derivative: f′(1) = 3(1) + 2.
Step 4: To find the slope of the tangent line at x = 2, substitute x = 2 into the derivative: f′(2) = 3(2) + 2.
Step 5: To find the slope of the tangent line at x = 3, substitute x = 3 into the derivative: f′(3) = 3(3) + 2.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is represented as f'(x) and provides the slope of the tangent line to the curve at that specific point. In this question, f'(x) = 3x + 2 indicates how the slope varies with x.
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05:44
Derivatives
Tangent Line
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 this line is equal to the derivative of the function at that point. To find the slope of the tangent line at x = 1, 2, and 3, we evaluate the derivative at these x-values.
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05:13Slopes of Tangent Lines
Evaluating Functions
Evaluating a function involves substituting a specific value into the function to find its output. In this context, we will substitute x = 1, 2, and 3 into the derivative f'(x) = 3x + 2 to find the slopes of the tangent lines at these points. This process is essential for determining how the function behaves at specific locations.
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4:26Evaluating Composed Functions
Related Practice
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Textbook Question
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Find the derivative of the following functions.
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