If f(x) = 2×, show that f(x + h) – f(x) = 2×( 2" -1). h Simplify your answers completely at each step. f(x) = 2X, so f(x + h) – f(x) - 2* - - h (2*%( )- 2* h II
If f(x) = 2×, show that f(x + h) – f(x) = 2×( 2" -1). h Simplify your answers completely at each step. f(x) = 2X, so f(x + h) – f(x) - 2* - - h (2*%( )- 2* h II
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![If \( f(x) = 2^x \), show that
\[
\frac{f(x+h) - f(x)}{h} = 2^x \left( \frac{2^h - 1}{h} \right).
\]
Simplify your answers completely at each step.
\( f(x) = 2^x \), so
\[
\frac{f(x+h) - f(x)}{h} = \frac{2^{x+h} - 2^x}{h}
\]
\[
= \frac{(2^x)(2^h) - 2^x}{h}
\]
This expression highlights the process of deriving the limit definition of a derivative for exponential functions. The steps guide the simplification from the difference quotient to a form that demonstrates the factorization of common elements.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2cd8142-e63e-4851-bc3e-120e6814a336%2F629b68e2-fc5c-44d8-8c28-49d09077679e%2Fel5c4ej_processed.png&w=3840&q=75)
Transcribed Image Text:If \( f(x) = 2^x \), show that
\[
\frac{f(x+h) - f(x)}{h} = 2^x \left( \frac{2^h - 1}{h} \right).
\]
Simplify your answers completely at each step.
\( f(x) = 2^x \), so
\[
\frac{f(x+h) - f(x)}{h} = \frac{2^{x+h} - 2^x}{h}
\]
\[
= \frac{(2^x)(2^h) - 2^x}{h}
\]
This expression highlights the process of deriving the limit definition of a derivative for exponential functions. The steps guide the simplification from the difference quotient to a form that demonstrates the factorization of common elements.
![**Problem Statement:**
If $5000 is borrowed at a rate of 5.75% interest per year, compounded quarterly, find the amount due at the end of the given number of years. (Round your answers to the nearest cent.)
**Questions:**
(a) 4 years
$ [Input Box] ✖
(b) 6 years
$ [Input Box] ✖
(c) 8 years
$ [Input Box] ✖
**Explanation:**
Given that the principal amount is $5000, with an annual interest rate of 5.75% compounded quarterly, we are asked to calculate the future amount for different periods of time: 4 years, 6 years, and 8 years.
Each option provides a text box for input, which shows an "✖" symbol indicating incorrect or incomplete input.
To solve, use the compound interest formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount ($5000).
- \( r \) is the annual interest rate (5.75% or 0.0575).
- \( n \) is the number of times that interest is compounded per year (quarterly, so 4).
- \( t \) is the time the money is invested for in years.
Substitute the values for each case to find the required amounts.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2cd8142-e63e-4851-bc3e-120e6814a336%2F629b68e2-fc5c-44d8-8c28-49d09077679e%2Frr3n8t_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
If $5000 is borrowed at a rate of 5.75% interest per year, compounded quarterly, find the amount due at the end of the given number of years. (Round your answers to the nearest cent.)
**Questions:**
(a) 4 years
$ [Input Box] ✖
(b) 6 years
$ [Input Box] ✖
(c) 8 years
$ [Input Box] ✖
**Explanation:**
Given that the principal amount is $5000, with an annual interest rate of 5.75% compounded quarterly, we are asked to calculate the future amount for different periods of time: 4 years, 6 years, and 8 years.
Each option provides a text box for input, which shows an "✖" symbol indicating incorrect or incomplete input.
To solve, use the compound interest formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount ($5000).
- \( r \) is the annual interest rate (5.75% or 0.0575).
- \( n \) is the number of times that interest is compounded per year (quarterly, so 4).
- \( t \) is the time the money is invested for in years.
Substitute the values for each case to find the required amounts.
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