Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Question

Answered step-by-step

Suppose that we don't have a formula for $ g(x) $ but we know that $ g(2) = -4 $ and $ g'(x) = \sqrt {x^2 + 5} $ for all $ x. $(a) Use a linear approximation to estimate $ g(1.95) $ and $ g(2.05). $(b) Are your estimate in part (a) too large or too small? Explain.

Video Answer

Solved by verified expert

This problem has been solved!

Try Numerade free for 7 days

Like

Report

Official textbook answer

Video by Amrita Bhasin

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 10

Linear Approximation and Differentials

Derivatives

Differentiation

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

02:59

$$ \begin{array}{l}{\text …

04:29

Suppose that we don't…

04:35

01:57

01:18

Suppose that the only info…

04:03

06:37

04:44

01:46

Okay. The first thing you know can do is we could substitute G prime of two into two squared plus five, which is a square to four plus five, which is scared of mine, which is equivalent to three. So now we have a G F ax is negative for plus three times X minus two, which is negative for post three acts minus six, which is three X minus 10. Therefore, we have G of 1.95 this three times 1.95 I was 10 negative. 4.15 Three times 2.5 minus tongue as negative. 3.85 Okay, we know that the second derivative is gonna be axe over the square root of ax squared. Plus five. Don't fit the chain role. No, we know the linear approximation is gonna be a line there for all the points away from the middle point to negative four from the center. X equals two are less than the actual values. Remember, G is calm cave up, so the estimates are less. Therefore, we haven't underestimate

View More Answers From This Book

Find Another Textbook