Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 2 - Limits

Problema 95a

Capitolo 2, Problema 95a

Sketch a graph of y=2^x and carefully draw three secant lines connecting the points P(0, 1) and Q(x,2^x), for x=−3,−2, and −1.

Guida verificata passo dopo passo

Step 1: Understand the function y=2^x. This is an exponential function where the base is 2. The graph of y=2^x is a curve that increases rapidly as x increases, and it passes through the point (0, 1) because 2^0 = 1.

Step 2: Plot the point P(0, 1) on the graph. This is a fixed point on the curve y=2^x.

Step 3: Calculate the y-values for the points Q(x, 2^x) when x = -3, -2, and -1. For x = -3, Q is (-3, 2^-3). For x = -2, Q is (-2, 2^-2). For x = -1, Q is (-1, 2^-1).

Step 4: Plot the points Q(-3, 2^-3), Q(-2, 2^-2), and Q(-1, 2^-1) on the graph. These points will be on the curve y=2^x.

Step 5: Draw secant lines connecting the point P(0, 1) to each of the points Q(-3, 2^-3), Q(-2, 2^-2), and Q(-1, 2^-1). These lines represent the average rate of change of the function between P and each Q.

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Exponential Functions

Exponential functions are mathematical expressions in the form y = a^x, where 'a' is a positive constant. In this case, y = 2^x represents an exponential function with a base of 2. These functions exhibit rapid growth as x increases and approach zero as x decreases, making them essential for understanding the behavior of the graph.

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Exponential Functions

Secant Lines

A secant line is a straight line that connects two points on a curve. In this context, the secant lines connect the fixed point P(0, 1) on the graph of y = 2^x to various points Q(x, 2^x) for specific values of x. The slope of the secant line provides an average rate of change of the function between the two points, which is crucial for analyzing the function's behavior.

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Percorso guidato

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Slopes of Tangent Lines

Graphing Techniques

Graphing techniques involve plotting points and understanding the shape and behavior of functions on a coordinate plane. For y = 2^x, it is important to accurately plot points such as P(0, 1) and Q(x, 2^x) for x = -3, -2, and -1. This helps visualize the exponential growth and the relationship between the points and the secant lines drawn between them.

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Percorso guidato

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Graphing The Derivative

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