Textbook Question
7–14. Find the derivative the following ways:
a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.
f(w) = w³ -w / w
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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Exponential Functions
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
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
21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(x) = 4x²+1; a= 2,4
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?
Textbook Question
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]