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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 77a
Chapter 3, Problem 77a

Graph the function f(x)={x        if x≤0x+1 if x>0f(x)=\(\begin{cases}\)x~~~~~~~~\(\text{if}\)~x\(\leq{0}\[\x\)+1~\(\text{if}\)~x\(\gt{0}\]\end{cases}\).

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Step 1: Understand the piecewise function. The function f(x) is defined in two parts: f(x) = x for x ≤ 0 and f(x) = x + 1 for x > 0.
Step 2: Graph the first part of the function, f(x) = x, for x ≤ 0. This is a straight line through the origin with a slope of 1, but only for x-values less than or equal to 0.
Step 3: Graph the second part of the function, f(x) = x + 1, for x > 0. This is a straight line with a slope of 1, starting at the point (0, 1) and continuing for x-values greater than 0.
Step 4: Identify the point of transition at x = 0. For x = 0, the function value is 0 from the first part, so the point (0, 0) is included in the graph.
Step 5: Combine the two parts on the same set of axes. Ensure the graph is continuous at x = 0, with a closed circle at (0, 0) and an open circle at (0, 1) to indicate the transition between the two parts.

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Key Concepts

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

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, the function f(x) has two cases: for x less than or equal to 0, it equals x, and for x greater than 0, it equals x + 1. Understanding how to evaluate and graph these distinct segments is crucial for visualizing the overall function.
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Piecewise Functions

Graphing Techniques

Graphing techniques involve plotting points and understanding the behavior of functions across different intervals. For piecewise functions, it is essential to identify the points where the function changes its definition, ensuring that the graph accurately reflects the function's behavior at those transition points.
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Graphing The Derivative

Continuity and Discontinuity

Continuity refers to a function being unbroken and having no gaps in its graph. In the case of the given piecewise function, it is important to check if the function is continuous at x = 0, where the definition changes. Analyzing limits and function values at this point helps determine if there is a jump or removable discontinuity.
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Intro to Continuity
Related Practice
Textbook Question

Derivatives from a graph Let F = f + g and G = 3f - g, where the graphs of f and g are shown in the figure. Find the following derivatives. <IMAGE>

F'(2)

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Textbook Question

For x > 0, what is f′(x)?

Textbook Question

{Use of Tech} The Witch of Agnesi The graph of y = a³ / x²+a², where a is a constant, is called the witch of Agnesi (named after the 18th-century Italian mathematician Maria Agnesi).

b. Plot the function and the tangent line found in part (a).

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Textbook Question

{Use of Tech} The Witch of Agnesi The graph of y = a3 / (x2 + a2), where a is a constant, is called the witch of Agnesi (named after the 18th-century Italian mathematician Maria Agnesi).

Let a = 3 and find an equation of the line tangent to y = 27 / (x2 + 9) at x = 2.

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Textbook Question

Derivatives from a table Use the following table to find the given derivatives. <IMAGE>

d/dx (f(x)g(x)) |x=1

Textbook Question

For x < 0, what is f′(x)?