Textbook Question
Find an equation of the line tangent to the given curve at a.
y = (x + 5) / (x - 1); a = 3
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 3.10.62a
62–65. {Use of Tech} Graphing f and f'
a. Graph f with a graphing utility.
f(x) = (x−1) sin^−1 x on [−1,1]
Verified step by step guidance1
Step 1: Understand the function f(x) = (x - 1) sin^−1(x). This function is composed of two parts: a linear term (x - 1) and the inverse sine function sin^−1(x). The domain of sin^−1(x) is [-1, 1], which matches the interval given for graphing.
Step 2: Use a graphing utility to plot the function f(x) over the interval [-1, 1]. Ensure that the graphing tool can handle inverse trigonometric functions and is set to the correct interval.
Step 3: Analyze the graph of f(x). Look for key features such as intercepts, maxima, minima, and points of inflection. Note how the linear term (x - 1) affects the shape of the graph in combination with sin^−1(x).
Step 4: To graph f'(x), first find the derivative of f(x). Use the product rule for differentiation, which states that if you have a function h(x) = u(x)v(x), then h'(x) = u'(x)v(x) + u(x)v'(x). Apply this to f(x) = (x - 1) sin^−1(x).
Step 5: After finding f'(x), use the graphing utility to plot the derivative over the same interval [-1, 1]. Compare the graphs of f(x) and f'(x) to understand how the derivative reflects the rate of change and behavior of the original function.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Graphing Functions
Graphing functions involves plotting points on a coordinate system to visualize the behavior of the function. For the function f(x) = (x−1) sin^−1 x, understanding its domain and range is crucial, especially since it is defined on the interval [-1, 1]. A graphing utility can help illustrate key features such as intercepts, maxima, minima, and asymptotic behavior.
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Graph of Sine and Cosine Function
Inverse Sine Function
The inverse sine function, denoted as sin^−1 x or arcsin x, is the function that returns the angle whose sine is x. Its range is limited to [-π/2, π/2], which is important when analyzing the function f(x) = (x−1) sin^−1 x. Understanding how this function behaves within its domain helps in predicting the overall shape of f.
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Derivative and Its Graph
The derivative of a function, denoted as f', represents the rate of change of the function with respect to its variable. Graphing f' provides insights into the function's increasing or decreasing behavior, as well as its critical points. For f(x) = (x−1) sin^−1 x, calculating f' will reveal where the function has local maxima or minima, which is essential for a complete analysis.
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Related Practice
Textbook Question
13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
³√x+³√y⁴ = 2;(1,1)
1
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Textbook Question
Tracking a dive A biologist standing at the bottom of an 80-foot vertical cliff watches a peregrine falcon dive from the top of the cliff at a 45° angle from the horizontal (see figure). <IMAGE>
a. Express the angle of elevation θ from the biologist to the falcon as a function of the height h of the bird above the ground. (Hint: The vertical distance between the top of the cliff and the falcon is 80−h.)
Textbook Question
Airline travel The following figure shows the position function of an airliner on an out-and-back trip from Seattle to Minneapolis, where s = f(t) is the number of ground miles from Seattle t hours after take-off at 6:00 A.M. The plane returns to Seattle 8.5 hours later at 2:30 P.M. <IMAGE>
a. Calculate the average velocity of the airliner during the first 1.5 hours of the trip (0 ≤ t ≤ 1.5).
Textbook Question
62–65. {Use of Tech} Graphing f and f'
a. Graph f with a graphing utility.
f(x)=e^−x tan^−1 x on [0,∞)
Textbook Question
If two opposite sides of a rectangle increase in length, how must the other two opposite sides change if the area of the rectangle is to remain constant?