Textbook Question
Explain Rolle’s Theorem with a sketch.
Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 4, Problem 4.7.47
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_v→3 (v-1-√(v²-5)) / (v-3)
Verified step by step guidance1
First, substitute v = 3 into the limit expression to check if it results in an indeterminate form. You will find that both the numerator and denominator become zero, indicating a 0/0 indeterminate form.
Since the limit results in a 0/0 indeterminate form, l'Hôpital's Rule can be applied. This rule states that if the limit of f(v)/g(v) as v approaches a certain value results in 0/0 or ∞/∞, then the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately.
Differentiate the numerator, which is (v - 1 - √(v² - 5)). The derivative of v - 1 is 1, and the derivative of -√(v² - 5) is -1/(2√(v² - 5)) * (2v) using the chain rule.
Differentiate the denominator, which is simply v - 3. The derivative of v - 3 is 1.
Apply l'Hôpital's Rule by taking the limit of the new expression formed by the derivatives: lim_v→3 [1 - v/√(v² - 5)] / 1. Substitute v = 3 into this expression to evaluate the limit.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding the function's behavior near points of interest, including points where the function may not be explicitly defined. Evaluating limits is essential for determining continuity, derivatives, and integrals.
Recommended video:
05:50
One-Sided Limits
L'Hôpital's Rule
L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. The rule states that if the limit of a quotient of two functions yields an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately, then re-evaluating the limit. This technique simplifies the process of finding limits in complex expressions.
Recommended video:
Guided course
5:50Power Rules
Square Root Functions
Square root functions, such as √(v²-5), are important in calculus as they can introduce complexities in limit evaluations. Understanding how to manipulate and simplify expressions involving square roots is crucial, especially when approaching limits that may lead to indeterminate forms. Recognizing the behavior of square root functions near specific values helps in applying techniques like L'Hôpital's Rule effectively.
Recommended video:
7:24Multiplying & Dividing Functions
Related Practice
Textbook Question
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = x²/₃ (x²-4)
Textbook Question
{Use of Tech} Critical points and extreme values
a. Find the critical points of the following functions on the given interval. Use a root finder, if necessary.
b. Use a graphing utility to determine whether the critical points correspond to local maxima, local minima, or neither.
c. Find the absolute maximum and minimum values on the given interval, if they exist
h(x) (5-x)/(x² + 2x - 3) on [-10,10]
Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→0 (x + cos x)¹/ˣ
Textbook Question
Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.
p(x) = 3 sec² x
Textbook Question
Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition.
f(x) = 8x³ + sin x; F(0) = 2