Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. F(x) = x³ - 4x + 100 and G(x) = x³ - 4x - 100 are antiderivatives of the same function.
52
views
Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
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 4.4.8a
Sketch a graph of a function f with the following properties.
f' < 0 and f" < 0, for x < -1
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 < -1.
For x < -1, since f'(x) < 0, the slope of the tangent line to the graph of f is negative, meaning the graph is sloping downward.
For x < -1, since f''(x) < 0, the graph is concave down, meaning it curves downward like an upside-down bowl.
Combine the two properties: The graph of f for x < -1 should be a decreasing curve that is concave down. This means the graph slopes downward and becomes steeper as x decreases.
Sketch the graph: Start at x = -1 and draw a curve that decreases and bends downward as x moves to the left. Ensure the curve reflects both the decreasing nature (f'(x) < 0) and the concave down property (f''(x) < 0).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
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 of the function 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 < -1.
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 understanding how the graph behaves in the specified interval, indicating that the function is not only decreasing but also bending downwards.
Recommended video:
06:02The Second Derivative Test: Finding Local Extrema
Graph Behavior
The overall behavior of a graph is influenced by both the first and second derivatives. In this case, since f' < 0 and f'' < 0 for x < -1, the graph will show a downward slope that becomes steeper as x decreases. Recognizing this behavior helps in accurately sketching the function, ensuring it reflects the properties of being decreasing and concave down.
Recommended video:
06:15Graphing The Derivative
Related Practice
Textbook Question
Folded boxes
a. Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 5 ft by 8 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way.
319
views
Textbook Question
Rectangles beneath a line
a. A rectangle is constructed with one side on the positive x-axis, one side on the positive y-axis, and the vertex opposite the origin on the line y = 10 - 2x. What dimensions maximize the area of the rectangle? What is the maximum area?
212
views
Textbook Question
Maximizing profit Suppose a tour guide has a bus that holds a maximum of 100 people. Assume his profit (in dollars) for taking people on a city tour is P(n) = n(50 - 0.5n) - 100. (Although P is defined only for positive integers, treat it as a continuous function.)
a. How many people should the guide take on a tour to maximize the profit?
342
views
Textbook Question
Population models The population of a species is given by the function P(t) = Kt²/(t² + b) , where t ≥ 0 is measured in years and K and b are positive real numbers.
a. With K = 300 and b = 30, what is lim_t→∞ P(t), the carrying capacity of the population?
333
views
Textbook Question
Suppose the objective function P= xy is subject to the constraint 10x + y = 100, where x and y are real numbers.
a. Eliminate the variable y from the objective function so that P is expressed as a function of one variable x.
276
views