/*! 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 33 Use the limit definition to find... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 33

Use the limit definition to find the derivative of the function. $$ h(t)=\sqrt{t-1} $$

Short Answer

Expert verified
The derivative of the function \( h(t) = \sqrt{t - 1} \) is \( h'(t) = \frac{1}{2(t - 1)^{1/2}} \).

Step by step solution

01

Rewrite the function

Firstly, express the function in a more convenient form for differentiation. Here, \( h(t) = \sqrt{t - 1} \) can be rewritten as \( h(t) = (t - 1)^{1/2} \). It is easier to differentiate in this form.
02

Apply the limit definition

The limit definition of a derivative is \( h'(t) = \lim_{s\to t} \frac{h(s) - h(t)}{s - t} \). Apply this definition with the function \( h(t) = (t - 1)^{1/2} \). You get \( h'(t) = \lim_{s\to t} \frac{(s - 1)^{1/2} - (t - 1)^{1/2}}{s - t} \). Now the task is to simplify the right side of the equation.
03

Simplify the expression

Multiplying the numerator and denominator by the conjugate of the numerator will get rid of the fraction. The conjugate of the numerator is \( (s + 1)^{1/2} + (t - 1)^{1/2} \), and after multiplication, you get: \( h'(t) = \lim_{s\to t} \frac{(s - t)}{(s - t)[(s - 1)^{1/2} + (t - 1)^{1/2}]} = \frac{1}{2(t - 1)^{1/2}} \cdot (1 - 0) = \frac{1}{2(t - 1)^{1/2}} \)
04

Evaluating the limit

Now, simply evaluate the limit as \( s \to t \) which will lead to the final derivative.

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

The percent \(P\) of defective parts produced by a new employee \(t\) days after the employee starts work can be modeled by \(P=\frac{t+1750}{50(t+2)}\) Find the rates of change of \(P\) when (a) \(t=1\) and (b) \(t=10\).

Match the function with the rule that you would use to find the derivative most efficiently. (a) Simple Power Rule (b) Constant Rule (c) General Power Rule (d) Quotient Rule $$ f(x)=\frac{2}{1-x^{3}} $$

Match the function with the rule that you would use to find the derivative most efficiently. (a) Simple Power Rule (b) Constant Rule (c) General Power Rule (d) Quotient Rule $$ f(x)=\sqrt[3]{8^{2}} $$

Find an equation of the tangent line to the graph of \(f\) at the point \((2, f(2)) .\) Use a graphing utility to check your result by graphing the original function and the tangent line in the same viewing window. $$ f(x)=\left(4-3 x^{2}\right)^{-2 / 3} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=\sqrt{x}(x-2)^{2} $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

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.