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• Find f' (-1) for f(x) = (3x + 2)² in each of the following ways: • Create a table of secant slopes with h = .1, h =-.1, h = .05, and h =-.05 and use these to estimate; • Use the limit definition of the derivative; • Use the Power Rule (some algebra may be required) and other rules; • Use the Product or Quotient Rule. dy • Find for y=in each of the following ways: dic • Use the limit definition of the derivative; • Use the Power Rule (some algebra may be required) and other rules; • Use the Product or Quotient Rule. • Which of these did you prefer in each case? Why? 3 dy For y = algebraically find the domain of dx • How does this relate to the graph of y=3? (You can use technology to graph this.) . dy • How does this relate to the limit definition of ? Be specific about what limits dx do or do not exist to describe this.

• Find f' (-1) for f(x) = (3x + 2)² in each of the following ways: • Create a table of secant slopes with h = .1, h =-.1, h = .05, and h =-.05 and use these to estimate; • Use the limit definition of the derivative; • Use the Power Rule (some algebra may be required) and other rules; • Use the Product or Quotient Rule. dy • Find for y=in each of the following ways: dic • Use the limit definition of the derivative; • Use the Power Rule (some algebra may be required) and other rules; • Use the Product or Quotient Rule. • Which of these did you prefer in each case? Why? 3 dy For y = algebraically find the domain of dx • How does this relate to the graph of y=3? (You can use technology to graph this.) . dy • How does this relate to the limit definition of ? Be specific about what limits dx do or do not exist to describe this.

Calculus: Early Transcendentals
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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I need help with the bullet point starting with 'Find dydx for y=3/(sqrt x) in each...' and the three bulletpoints underneath it. Thank you!

**Calculus: Techniques and Applications** --- **Finding the Derivative** 1. **Calculate \( f'(-1) \) for \( f(x) = (3x + 2)^2 \) using different methods:** - **Create a table of secant slopes**: - Use \( h = 0.1 \), \( h = -0.1 \), \( h = 0.05 \), and \( h = -0.05 \) to estimate the slope. - **Use the limit definition of the derivative**: \[ f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h} \] - **Use the Power Rule** (algebraic manipulation may be required): \[ \left( u^n \right)' = n u^{n-1} \cdot \frac{du}{dx} \] - **Use the Product or Quotient Rule**: \[ (uv)' = u'v + uv' \quad \text{and} \quad \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \] 2. **Calculate \( \frac{dy}{dx} \) for \( y = \frac{3}{\sqrt{x}} \) using various methods:** - **Use the limit definition of the derivative**. - **Use the Power Rule** (some algebraic manipulation may be required): \[ y = 3x^{-\frac{1}{2}} \text{ and } \frac{d}{dx} \left( x^n \right) = nx^{n-1} \] - **Use the Product or Quotient Rule**. 3. **Preference Analysis**: - Reflect on which method you preferred in each case and explain why. **Domain and Graph Analysis** 4. **For \( y = \frac{3}{\sqrt{x}} \), find the domain of \( \frac{dy}{dx} \) algebraically.** - **Graphical Analysis**: Explore how the domain of the derivative relates to the graph of \( y = \frac{3}{\sqrt{x}} \).
Transcribed Image Text:**Calculus: Techniques and Applications** --- **Finding the Derivative** 1. **Calculate \( f'(-1) \) for \( f(x) = (3x + 2)^2 \) using different methods:** - **Create a table of secant slopes**: - Use \( h = 0.1 \), \( h = -0.1 \), \( h = 0.05 \), and \( h = -0.05 \) to estimate the slope. - **Use the limit definition of the derivative**: \[ f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h} \] - **Use the Power Rule** (algebraic manipulation may be required): \[ \left( u^n \right)' = n u^{n-1} \cdot \frac{du}{dx} \] - **Use the Product or Quotient Rule**: \[ (uv)' = u'v + uv' \quad \text{and} \quad \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \] 2. **Calculate \( \frac{dy}{dx} \) for \( y = \frac{3}{\sqrt{x}} \) using various methods:** - **Use the limit definition of the derivative**. - **Use the Power Rule** (some algebraic manipulation may be required): \[ y = 3x^{-\frac{1}{2}} \text{ and } \frac{d}{dx} \left( x^n \right) = nx^{n-1} \] - **Use the Product or Quotient Rule**. 3. **Preference Analysis**: - Reflect on which method you preferred in each case and explain why. **Domain and Graph Analysis** 4. **For \( y = \frac{3}{\sqrt{x}} \), find the domain of \( \frac{dy}{dx} \) algebraically.** - **Graphical Analysis**: Explore how the domain of the derivative relates to the graph of \( y = \frac{3}{\sqrt{x}} \).
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