Textbook Question
21–30. Derivatives
b. Evaluate f'(a) for the given values of a.
f(x) = 4x²+1; a= 2,4
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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 3.3.74b
Suppose f(3) = 1 and f′(3) = 4. Let g(x) = x2 + f(x) and h(x) = 3f(x).
Find an equation of the line tangent to y = h(x) at x = 3.
Verified step by step guidance1
Step 1: Identify the function h(x) and its derivative. We have h(x) = 3f(x). To find the derivative h'(x), use the constant multiple rule: h'(x) = 3f'(x).
Step 2: Evaluate h(x) and h'(x) at x = 3. We know f(3) = 1 and f'(3) = 4. Therefore, h(3) = 3f(3) = 3 * 1 = 3 and h'(3) = 3f'(3) = 3 * 4 = 12.
Step 3: Use the point-slope form of the equation of a line to find the tangent line. The point-slope form is y - y_1 = m(x - x_1), where m is the slope and (x_1, y_1) is a point on the line.
Step 4: Substitute the values into the point-slope form. Here, the slope m is h'(3) = 12, and the point (x_1, y_1) is (3, h(3)) = (3, 3).
Step 5: Write the equation of the tangent line using the point-slope form: y - 3 = 12(x - 3).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 the tangent line is given by the derivative of the function at that point. In this case, to find the tangent line to h(x) at x = 3, we need to calculate h'(3) to determine the slope.
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Slopes of Tangent Lines
Derivative
The derivative of a function measures how the function's output changes as its input changes. It is a fundamental concept in calculus that provides the slope of the tangent line at any point on the curve. For the function h(x) = 3f(x), we will use the chain rule to find h'(x) and evaluate it at x = 3.
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05:44Derivatives
Function Composition
Function composition involves combining two functions where the output of one function becomes the input of another. In this problem, g(x) and h(x) are defined in terms of f(x). Understanding how to differentiate composite functions is essential for finding the derivative of h(x) and subsequently the equation of the tangent line.
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Related Practice
Textbook Question
79–82. {Use of Tech} Visualizing tangent and normal lines
b. Graph the tangent and normal lines on the given graph.
x⁴ = 2x²+2y²; (x0, y0)=(2, 2) (kampyle of Eudoxus)
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Textbook Question
Use a graphing utility to plot the curve and the tangent line.
y = cos x / 1−cos x; x = π/3
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Textbook Question
Surface area of a cone The lateral surface area of a cone of radius r and height h (the surface area excluding the base) is A = πr√r²+h².
b. Evaluate this derivative when r=30 and h=40.
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Textbook Question
Volume of a torus The volume of a torus (doughnut or bagel) with an inner radius of a and an outer radius of b is V=π²(b+a)(b−a)²/4.
b. Evaluate this derivative when a=6 and b=10.
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Textbook Question
Vertical tangent lines
b. Does the curve have any horizontal tangent lines? Explain.
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