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 92b

Chapter 2, Problem 92b

Suppose you park your car at a trailhead in a national park and begin a 2-hr hike to a lake at 7 A.M. on a Friday morning. On Sunday morning, you leave the lake at 7 A.M. and start the 2-hr hike back to your car. Assume the lake is 3 mi from your car. Let f(t) be your distance from the car t hours after 7 a.m. on Friday morning, and let g(t) be your distance from the car t hours after 7 a.m. on Sunday morning.

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

Verified step by step guidance

Start by understanding the functions: f(t) represents your distance from the car on Friday, and g(t) represents your distance from the car on Sunday.

Define h(t) = f(t) - g(t). This function represents the difference in your distance from the car between Friday and Sunday at the same time t.

To find h(0), evaluate f(0) and g(0). Since you start at the car both times, f(0) = 0 and g(0) = 0. Therefore, h(0) = f(0) - g(0).

To find h(2), evaluate f(2) and g(2). Since the hike is 2 hours and the lake is 3 miles away, f(2) = 3 and g(2) = 3. Therefore, h(2) = f(2) - g(2).

Conclude by noting that h(t) represents the difference in your position relative to the car at the same time on Friday and Sunday.

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Key Concepts

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

Distance Functions

In this context, f(t) and g(t) represent the distance from the car at different times during the hike. f(t) describes the distance as you hike away from the car towards the lake, while g(t) describes the distance as you return to the car. Understanding how these functions behave over time is crucial for analyzing the overall journey.

Function Evaluation

Evaluating a function at a specific point involves substituting a value into the function to find the corresponding output. For h(t) = f(t) - g(t), finding h(0) and h(2) requires calculating f(0), g(0), f(2), and g(2). This process is essential for determining the differences in distance at those specific times.

Difference of Functions

The function h(t) = f(t) - g(t) represents the difference in distance from the car at any given time t. This concept is important for understanding how the distances from the car change over the course of the hike. Analyzing h(t) helps to identify the relationship between the two journeys and can reveal insights about the overall trip.

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Related Practice

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Let g(x)= {1 if x≥0

−1 if x<0.

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

Textbook Question

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.

Textbook Question

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

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

Textbook Question

Suppose you park your car at a trailhead in a national park and begin a 2-hr hike to a lake at 7 A.M. on a Friday morning. On Sunday morning, you leave the lake at 7 A.M. and start the 2-hr hike back to your car. Assume the lake is 3 mi from your car. Let f(t) be your distance from the car t hours after 7 a.m. on Friday morning, and let g(t) be your distance from the car t hours after 7 a.m. on Sunday morning.

a. Evaluate f(0), f(2), g(0), and g(2).

Textbook Question

Calculate the following limits using the factorization formula x^n−a^n=(x−a)(x^n−1+ax^n−2+a^2x^n−3+⋯+a^n−2x+a^n−1), where n is a positive integer and a is a real number.

lim x→1 x^6 − 1 / x − 1

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Textbook Question

Suppose g(x) = {x^2−5x if x≤−1

ax^3−7 if x>−1.

Determine a value of the constant a for which lim x→−1 g(x) exists and state the value of the limit, if possible.