Functions of the form f(x) x", where n = 1, 2, 3, ..., are often called power functions. The first two questions below revisit work we did earlier in Chapter 1, and the following questions extend those ideas to higher powers of x. Use the limit definition of the derivative to find f'(x) for f(x) = x². = Use the limit definition of the derivative to find f'(x) for f(x) = x³. Use the limit definition of the derivative to find f'(x) for f(x) = x²ª. Hint What do you conjecture is the derivative of f(x) = x¹³? Hint Based on your work in the previous questions, what do you conjecture is the derivative of f(x) = x³? Conjecture a formula for the derivative to find f(x) =x" that holds for any positive integer n. That is, given f(x) = x¹ where n is a positive integer, what do you think is the formula for f'(x)?
Functions of the form f(x) x", where n = 1, 2, 3, ..., are often called power functions. The first two questions below revisit work we did earlier in Chapter 1, and the following questions extend those ideas to higher powers of x. Use the limit definition of the derivative to find f'(x) for f(x) = x². = Use the limit definition of the derivative to find f'(x) for f(x) = x³. Use the limit definition of the derivative to find f'(x) for f(x) = x²ª. Hint What do you conjecture is the derivative of f(x) = x¹³? Hint Based on your work in the previous questions, what do you conjecture is the derivative of f(x) = x³? Conjecture a formula for the derivative to find f(x) =x" that holds for any positive integer n. That is, given f(x) = x¹ where n is a positive integer, what do you think is the formula for f'(x)?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Functions of the form f(x) = x", where n = 1, 2, 3, ..., are often called power functions. The first two
questions below revisit work we did earlier in Chapter 1, and the following questions extend those ideas to
higher powers of x.
Use the limit definition of the derivative to find f'(x) for f(x) = x².
Use the limit definition of the derivative to find f'(x) for f(x) = x³.
Use the limit definition of the derivative to find f'(x) for f(x) = x².
Hint
What do you conjecture is the derivative of f(x) = x¹³?
Hint
Based on your work in the previous questions, what do you conjecture is the derivative of f(x) = x³?
Conjecture a formula for the derivative to find f(x) = x" that holds for any positive integer n. That is,
given f(x)=x" where n is a positive integer, what do you think is the formula for f'(x)?
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