Textbook Question
Vertical tangent lines
a. Determine the points where the curve x+y³−y=1 has a vertical tangent line (see Exercise 60).
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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 3.10.65a
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,∞)
Verified step by step guidance1
Step 1: Understand the function f(x) = e^(-x) * tan^(-1)(x). This function is a product of the exponential function e^(-x) and the inverse tangent function tan^(-1)(x).
Step 2: Use a graphing utility or software that supports graphing, such as Desmos, GeoGebra, or a graphing calculator, to input the function f(x) = e^(-x) * tan^(-1)(x).
Step 3: Set the domain for the graph. Since the problem specifies the interval [0, ∞), ensure that the graphing utility is set to display the graph starting from x = 0 and extending towards positive infinity.
Step 4: Observe the behavior of the graph. Note how the exponential decay of e^(-x) affects the overall shape of the graph, and how the inverse tangent function, which approaches π/2 as x approaches infinity, influences the graph.
Step 5: Analyze the graph for key features such as intercepts, asymptotic behavior, and any points of interest. This will help in understanding the overall behavior of the function on the given interval.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Functions
Exponential functions are mathematical expressions in the form f(x) = a * e^(bx), where 'e' is Euler's number (approximately 2.718). In the given function f(x) = e^(-x) * tan^(-1)(x), the term e^(-x) represents a decaying exponential function, which approaches zero as x increases. Understanding the behavior of exponential functions is crucial for analyzing the overall shape and limits of the graph.
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Inverse Tangent Function
The inverse tangent function, denoted as tan^(-1)(x) or arctan(x), is the function that returns the angle whose tangent is x. It has a range of (-π/2, π/2) and approaches these limits as x approaches ±∞. In the context of the function f(x) = e^(-x) * tan^(-1)(x), this function influences the growth of f(x) as x increases, particularly since tan^(-1)(x) approaches π/2.
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Graphing Utilities
Graphing utilities are software or tools that allow users to visualize mathematical functions and their derivatives. They can plot complex functions, helping to analyze their behavior over specified intervals. For the function f(x) = e^(-x) * tan^(-1)(x) on the interval [0, ∞), using a graphing utility will provide insights into the function's growth, decay, and asymptotic behavior, which is essential for understanding its overall characteristics.
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Related Practice
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)
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Textbook Question
Use the definition of the derivative to determine d/dx(ax²+bx+c), where a, b, and c are constants.
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Textbook Question
Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = 1/ √x; a= 1/4
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Textbook Question
79–82. {Use of Tech} Visualizing tangent and normal lines <IMAGE>
a. Determine an equation of the tangent line and the normal line at the given point (x0, y0) on the following curves. (See instructions for Exercises 73–78.)
(x²+y²)² = 25/3 (x²-y²); (x0,y0) = (2,-1) (lemniscate of Bernoulli)
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Textbook Question
The volume V of a sphere of radius r changes over time t.
a. Find an equation relating dV/dt to dr/dt.
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