Textbook Question
Calculate the derivative of the following functions.
g(x) = x / e3x
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 52a
An angler hooks a trout and begins turning her circular reel at 1.5 rev/s. Assume the radius of the reel (and the fishing line on it) is 2 inches.
a. Let R equal the number of revolutions the angler has turned her reel and suppose L is the amount of line that she has reeled in. Find an equation for L as a function of R.
Verified step by step guidance1
Understand that the problem involves a circular motion where the reel is turning, and we need to relate the number of revolutions (R) to the length of the line reeled in (L).
Recognize that each complete revolution of the reel corresponds to the circumference of the circle formed by the reel. The circumference C of a circle is given by the formula: C = 2πr, where r is the radius.
Substitute the given radius of the reel into the circumference formula. Here, the radius r is 2 inches, so the circumference C = 2π(2) = 4π inches.
Since each revolution reels in a length of line equal to the circumference of the reel, the amount of line L reeled in after R revolutions is L = R * C.
Substitute the expression for the circumference into the equation for L: L = R * 4π. Therefore, the equation for L as a function of R is L(R) = 4πR.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Circumference of a Circle
The circumference of a circle is the distance around it, calculated using the formula C = 2πr, where r is the radius. In this context, the radius of the reel is 2 inches, so the circumference represents the length of fishing line that is reeled in with each complete revolution of the reel.
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Linear Relationship
A linear relationship describes how one variable changes in relation to another. In this case, the amount of line L that is reeled in is directly proportional to the number of revolutions R made by the reel, which can be expressed as L = C * R, where C is the circumference of the reel.
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Rate of Revolution
The rate of revolution indicates how quickly the reel is turned, measured in revolutions per second (rev/s). Here, the angler turns the reel at a rate of 1.5 rev/s, which helps determine how fast the line is being reeled in over time, linking the concepts of time, revolutions, and the length of line.
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Related Practice
Textbook Question
Where is the function continuous? Differentiable? Use the graph of f in the figure to do the following. <IMAGE>
b. Find the values of x in (0, 3) at which f is not differentiable.
Textbook Question
Calculate the derivative of the following functions.
y = sin(sin(ex))
1
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Textbook Question
{Use of Tech} A different interpretation of marginal cost Suppose a large company makes 25,000 gadgets per year in batches of x items at a time. After analyzing setup costs to produce each batch and taking into account storage costs, planners have determined that the total cost C(x) of producing 25,000 gadgets in batches of x items at a time is given by C(x) = 1,250,000+125,000,000 / x + 1.5x.
a. Determine the marginal cost and average cost functions. Graph and interpret these functions.
Textbook Question
The position (in meters) of a marble, given an initial velocity and rolling up a long incline, is given by s = 100t / t+1, where t is measured in seconds and s=0 is the starting point.
b. Find the velocity function for the marble.
Textbook Question
Where is the function continuous? Differentiable? Use the graph of f in the figure to do the following. <IMAGE>
a. Find the values of x in (0, 3) at which f is not continuous.