Textbook Question
Find an equation of the line tangent to the following curves at the given value of x.
y = 4 sin x cos x; x = π/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 3.10.65b
62–65. {Use of Tech} Graphing f and f'
b. Compute and graph f'.
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 two functions: e^(-x) and tan^(-1)(x). To find the derivative f'(x), we will use the product rule.
Step 2: Recall the product rule for derivatives, which states that if you have two functions u(x) and v(x), then the derivative of their product is given by (uv)' = u'v + uv'. Here, let u(x) = e^(-x) and v(x) = tan^(-1)(x).
Step 3: Compute the derivative of u(x) = e^(-x). The derivative u'(x) is found using the chain rule: u'(x) = -e^(-x).
Step 4: Compute the derivative of v(x) = tan^(-1)(x). The derivative v'(x) is 1/(1 + x^2).
Step 5: Apply the product rule to find f'(x): f'(x) = u'(x)v(x) + u(x)v'(x) = (-e^(-x))tan^(-1)(x) + e^(-x)/(1 + x^2). Now, use a graphing tool to plot both f(x) and f'(x) over the interval [0, ∞).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function measures how the function's output changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. For the function f(x) = e^(-x) * arctan(x), the derivative f'(x) can be computed using the product rule and chain rule of differentiation.
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Derivatives
Graphing Functions
Graphing a function involves plotting its output values against its input values on a coordinate system. For the function f(x) = e^(-x) * arctan(x), understanding its behavior as x approaches 0 and infinity is crucial. The graph of f' will provide insights into the function's increasing or decreasing behavior, as well as its critical points.
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5:53Graph of Sine and Cosine Function
Behavior at Infinity
Analyzing the behavior of a function as x approaches infinity helps determine its long-term trends. For f(x) = e^(-x) * arctan(x), as x increases, e^(-x) approaches 0, while arctan(x) approaches π/2. This interplay affects the overall behavior of f and its derivative f', which is essential for understanding the function's limits and asymptotic behavior.
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03:07Cases Where Limits Do Not Exist
Related Practice
Textbook Question
An object oscillates along a vertical line, and its position in centimeters is given by y(t) = 30(sint - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.
Find the velocity of the oscillator, v(t) =y′(t).
Textbook Question
{Use of Tech} Hours of daylight The number of hours of daylight at any point on Earth fluctuates throughout the year. In the Northern Hemisphere, the shortest day is on the winter solstice and the longest day is on the summer solstice. At 40° north latitude, the length of a day is approximated by D(t) = 12−3 cos (2π(t+10) / 365), where D is measured in hours and 0≤t≤365 is measured in days, with t=0 corresponding to January 1.
b. Find the rate at which the daylight function changes.
Textbook Question
21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(x) = 1/x+1; a = -1/2;5
Textbook Question
60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>
a. Find equations of all lines tangent to the curve at the given value of x.
x+y³−y=1; x=1
Textbook Question
Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>
b. Find equations of all lines tangent to the curve y(x²+4)=8 when y=1.