Textbook Question
Optimal soda can
a. Classical problem Find the radius and height of a cylindrical soda can with a volume of 354 cm³ that minimize the surface area.
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Ch. 4 - Applications of the Derivative
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 4, Problem 4.4.7a
Sketch a graph of a function f with the following properties.
f' < 0 and f" < 0, for x < 3
Verified step by step guidance1
Understand the given conditions: f'(x) < 0 implies the function f is decreasing, and f''(x) < 0 implies the function is concave down for x < 3.
Start by sketching the graph for x < 3. Since f'(x) < 0, the graph should slope downward as x approaches 3. This indicates a decreasing function.
Next, incorporate the concavity condition f''(x) < 0. Concave down means the graph curves downward, resembling a 'frown' shape. Ensure the graph bends downward as it decreases.
Choose a point at x = 3 to mark the boundary where the given conditions apply. The behavior of the graph beyond x = 3 is not specified, so focus only on the region x < 3.
Label the axes and ensure the graph visually represents both decreasing behavior and concave down curvature for x < 3. This completes the sketch based on the given properties.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
First Derivative Test
The first derivative of a function, denoted as f', indicates the slope of the tangent line to the graph at any point. If f' < 0, the function is decreasing in that interval. Understanding this concept is crucial for sketching the graph, as it informs us that the function is moving downward for x < 3.
Recommended video:
07:09
The First Derivative Test: Finding Local Extrema
Second Derivative Test
The second derivative, denoted as f'', provides information about the concavity of the function. If f'' < 0, the function is concave down, meaning that the slope of the tangent line is decreasing. This concept is essential for determining the curvature of the graph, indicating that the function is bending downwards for x < 3.
Recommended video:
06:02The Second Derivative Test: Finding Local Extrema
Graph Behavior
Understanding how the first and second derivatives affect the graph's behavior is key to sketching it accurately. With f' < 0 and f'' < 0 for x < 3, the graph will not only be decreasing but also concave down, suggesting a continuous downward slope that becomes steeper as x approaches 3. This overall behavior shapes the visual representation of the function.
Recommended video:
06:15Graphing The Derivative
Related Practice
Textbook Question
Values of related functions Suppose f is differentiable on (-∞,∞) and assume it has a local extreme value at the point x = 2, where f(2) = 0. Let g(x) = xf(x) + 1 and let h(x) = xf(x) + x +1, for all values of x.
a. Evaluate g(2), h(2), g'(2), and h'(2).
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Textbook Question
Do dogs know calculus? A mathematician stands on a beach with his dog at point A. He throws a tennis ball so that it hits the water at point B. The dog, wanting to get to the tennis ball as quickly as possible, runs along the straight beach line to point D and then swims from point D to point B to retrieve his ball. Assume C is the point on the edge of the beach closest to the tennis ball (see figure). <IMAGE>
a. Assume the dog runs at speed r and swims at speed s, where r > s and both are measured in meters per second. Also assume the lengths of BC, CD, and AC are x, y, and z, respectively. Find a function T(y) representing the total time it takes for the dog to get to the ball.
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Textbook Question
{Use of Tech} Investment problem A one-time investment of \(2500 is deposited in a 5-year savings account paying a fixed annual interest rate r, with monthly compounding. The amount of money in the account after 5 years is a(r) = 2500(1 + r/12)⁶⁰.
a. Use Newton’s method to find the value of r if the goal is to have \)3200 in the account after 5 years.
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Textbook Question
107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .
a. Find the velocity of the object for all relevant times.
A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.
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Textbook Question
{Use of Tech} Fixed points of quadratics and quartics Let f(x) = ax(1 -x), where a is a real number and 0 ≤ a ≤ 1. Recall that the fixed point of a function is a value of x such that f(x) = x (Exercises 48–51).
a. Without using a calculator, find the values of a, with 0 ≤ a ≤ 4, such that f has a fixed point. Give the fixed point in terms of a.
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