Skip to main content
Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.4.11
Chapter 4, Problem 4.4.11

Sketch a continuous function f on some interval that has the properties described. Answers will vary.


The function f satisfies f'(-2) = 2, f'(0) = 0, f'(1) = -3 and f'(4) = 1.

Verified step by step guidance
1
Understand that f'(x) represents the derivative of the function f(x), which gives the slope of the tangent line to the curve at any point x. The given values of f'(x) indicate the slope of the function at specific points.
At x = -2, f'(-2) = 2, which means the slope of the tangent line to the function at x = -2 is 2. This suggests that the function is increasing at this point.
At x = 0, f'(0) = 0, indicating that the slope of the tangent line is zero. This typically means there is a horizontal tangent line at this point, suggesting a local maximum or minimum.
At x = 1, f'(1) = -3, which means the slope of the tangent line is -3. This indicates that the function is decreasing at this point.
At x = 4, f'(4) = 1, indicating that the slope of the tangent line is 1. This suggests that the function is increasing at this point. Use these slopes to sketch a function that transitions smoothly through these points, ensuring continuity.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is represented as f'(x) and provides information about the function's slope at that specific point. Understanding the values of the derivative at given points helps in determining the function's behavior, such as increasing or decreasing intervals.
Recommended video:
05:44
Derivatives

Critical Points

Critical points occur where the derivative of a function is zero or undefined. These points are significant because they can indicate local maxima, minima, or points of inflection. In the context of the given question, f'(0) = 0 suggests a critical point, which is essential for sketching the function and understanding its overall shape.
Recommended video:
04:50
Critical Points

Continuity

A function is continuous on an interval if there are no breaks, jumps, or holes in its graph. This property ensures that the function can be drawn without lifting the pencil from the paper. For the function f in the question, being continuous means that it must smoothly connect the points defined by the derivative values, influencing how the function is sketched across the specified interval.
Recommended video:
05:34
Intro to Continuity
Related Practice
Textbook Question

{Use of Tech} Fixed points An important question about many functions concerns the existence and location of fixed points. A fixed point of f is a value of x that satisfies the equation f(x) = x; it corresponds to a point at which the graph of f intersects the line y = x. Find all the fixed points of the following functions. Use preliminary analysis and graphing to determine good initial approximations.


f(x) = cos x

241
views
Textbook Question

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.


f(x) = sin² x

198
views
Textbook Question

Suppose f is differentiable on (-∞,∞) and f(5.01) - f(5) = 0.25.Use linear approximation to estimate the value of f'(5).

249
views
Textbook Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→1⁻ (1-x) tan πx/2

242
views
Textbook Question

Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.


f(x) = 2x⁻³ - x⁻²

196
views
Textbook Question

Suppose the position of an object moving horizontally after seconds is given by the function s(t) = 32t - t⁴, where 0 ≤ t ≤ 3 and s is measured in feet, with s > 0 corresponding to positions to the right of the origin. When is the object farthest to the right?

288
views