/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 75 Find the values of \(a\) and \(b... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 75

Find the values of \(a\) and \(b\) such that \(\lim _{x \rightarrow 0} \frac{a-\cos b x}{x^{2}}=2\).

Short Answer

Expert verified
The values of \(a\) and \(b\) are 0 and 卤2 respectively.

Step by step solution

01

Analyze the equation

The given equation is \(\lim _{x \rightarrow 0} \frac{a-\cos bx}{x^{2}}=2\). This scenario is an indeterminate form and it is required to determine the values of \(a\) and \(b\). L'Hopital's rule will need to be used to solve the problem.
02

Apply L'Hopital's Rule

According to L'Hopital鈥檚 rule, when the limit of a function as \(x\) approaches a certain value is indeterminate (i.e., either \(0/0\) or \(\infty/ \infty)\), the required limit can be found by computing the limit of the derivative of the numerator and denominator. The derivative of \(a\) is 0 as it's a constant, and the derivative of \(-cos(bx)\) is \(bsin(bx)\), while the derivative of \(x^2\) is \(2x\). Thus, \(\lim _{x \rightarrow 0} \frac{a-\cos bx}{x^{2}}\) = \(\lim _{x \rightarrow 0} \frac{0 + bsin(bx)}{2x}\).
03

Simplify the equation

We again use L'Hopital's rule to find the limit as x goes to 0 for the equation \(\lim _{x \rightarrow 0} \frac{bsin(bx)}{2x}\). The derivative of \(bsin(bx)\) is \(b^2cos(bx)\) and derivative of \(2x\) is 2. Thus, the equation simplifies to \(\lim _{x \rightarrow 0} \frac{b^2cos(bx)}{2}\). This gives the result as \(b^2/2\).
04

Equate to 2

Since we know from step 3 that our limit returns \(b^2/2\), we can equate this to 2 (the result of our limit as defined in the problem). Solve the equation \(b^2/2 = 2\). Solve for \(b\) to get \(b = 卤2\).
05

Find the Value of a

Substitute the value of \(b\) into \(\lim _{x \rightarrow 0} \frac{a-\cos bx}{x^{2}}=2\) to find the value of \(a\). Since the \(b\)\(-cos(bx)\) term will go to 0 as \(x\) approaches 0, the equation can be rewritten as follows: \(\lim _{x \rightarrow 0} \frac{a}{x^{2}}=2\). Simplifying this, we find that \(a = 0\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=\tan x, \quad y=0, \quad x=-\pi / 4, \quad x=\pi / 4 $$

Find the integral. Use a computer algebra system to confirm your result. $$ \int \cot ^{3} 2 x d x $$

Prove the following generalization of the Mean Value Theorem. If \(f\) is twice differentiable on the closed interval \([a, b],\) then \(f(b)-f(a)=f^{\prime}(a)(b-a)-\int_{a}^{b} f^{\prime \prime}(t)(t-b) d t\).

Rewrite the improper integral as a proper integral using the given \(u\) -substitution. Then use the Trapezoidal Rule with \(n=5\) to approximate the integral. $$ \int_{0}^{1} \frac{\sin x}{\sqrt{x}} d x, \quad u=\sqrt{x} $$

Sketch the graph of the hypocycloid of four cusps \(x^{2 / 3}+y^{2 / 3}=4\) and find its perimeter
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.