Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 2 - Limits

Problem 95a

Chapter 2, Problem 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.

Verified step by step guidance

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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Key Concepts

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

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.

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.

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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Textbook Question

Let g(x)= {1 if x≥0

−1 if x<0.

a. Write a formula for |g(x)|.

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b. Let h(t)=f(t)−g(t). Find h(0) and h(2).

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Find the horizontal asymptotes of each function using limits at infinity.

f(x) = (2e^x + 3) / (e^x + 1)

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Textbook Question

Find the horizontal asymptotes of each function using limits at infinity.

f(x) = (3e^5x + 7e^6x) / (9e^5x + 14e^6x)