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he product and quotient rules Activity 2.3.4. Use relevant derivative rules to answer each of the questions below. Throughout, be sure to use proper notation and carefully label any derivative you find by name. a. Let f(r) = (5r³+ sin(r))(4" – 2 cos(r)). Find f'(r). %3D cos(t) b. Let p(t) = Find p'(t). t6.6t c. Let g(z) = 3z"e² – 2z² sin(z) + Find g'(z). %3D d. A moving particle has its position in feet at time t in seconds given by the function s(t) = 2COSCS the particle's instantaneous velocity at the moment t = 1. 3 cos(t)-sin(t) Find e. Suppose that f(x) and g(x) are differentiable functions and it is known that f(3) = -2, f'(3) = 7, g(3) = 4, and gʻ(3) = –1. If p(x) = f(x) · g(x) and q(x): f(x) a(x) calculate p'ʻ(3) and q'(3). %3D %3D

he product and quotient rules Activity 2.3.4. Use relevant derivative rules to answer each of the questions below. Throughout, be sure to use proper notation and carefully label any derivative you find by name. a. Let f(r) = (5r³+ sin(r))(4" – 2 cos(r)). Find f'(r). %3D cos(t) b. Let p(t) = Find p'(t). t6.6t c. Let g(z) = 3z"e² – 2z² sin(z) + Find g'(z). %3D d. A moving particle has its position in feet at time t in seconds given by the function s(t) = 2COSCS the particle's instantaneous velocity at the moment t = 1. 3 cos(t)-sin(t) Find e. Suppose that f(x) and g(x) are differentiable functions and it is known that f(3) = -2, f'(3) = 7, g(3) = 4, and gʻ(3) = –1. If p(x) = f(x) · g(x) and q(x): f(x) a(x) calculate p'ʻ(3) and q'(3). %3D %3D

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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**Activity 2.3.4: Understanding Derivatives** In this activity, we will apply derivative rules to solve various problems. Ensure that you use proper notation and clearly label any derivatives you derive. a. **Function**: \( f(r) = (5r^3 + \sin(r))(4r - 2\cos(r)) \). **Task**: Determine \( f'(r) \). b. **Function**: \( p(t) = \frac{\cos(t)}{t^6 \cdot 6t} \). **Task**: Find \( p'(t) \). c. **Function**: \( q(z) = 3z^7e^z - 2z^2 \sin(z) + \frac{z}{z^2 + 1} \). **Task**: Calculate \( g'(z) \). d. **Scenario**: A moving particle’s position, in feet, at time \( t \) seconds is given by \( s(t) = \frac{3\cos(t) - \sin(t)}{t^4} \). **Task**: Find the particle’s instantaneous velocity at \( t = 1 \). e. **Given Data**: Assume \( f(x) \) and \( g(x) \) are differentiable functions with known values: - \( f(3) = -2 \) - \( f'(3) = 7 \) - \( g(3) = 4 \) - \( g'(3) = -1 \) - **Function Definitions**: - \( p(x) = f(x) \cdot g(x) \) - \( q(x) = \frac{f(x)}{g(x)} \) **Task**: Calculate \( p'(3) \) and \( q'(3) \) using the provided values. Through these exercises, you will enhance your understanding of derivative applications and strengthen your calculus skills.
Transcribed Image Text:**Activity 2.3.4: Understanding Derivatives** In this activity, we will apply derivative rules to solve various problems. Ensure that you use proper notation and clearly label any derivatives you derive. a. **Function**: \( f(r) = (5r^3 + \sin(r))(4r - 2\cos(r)) \). **Task**: Determine \( f'(r) \). b. **Function**: \( p(t) = \frac{\cos(t)}{t^6 \cdot 6t} \). **Task**: Find \( p'(t) \). c. **Function**: \( q(z) = 3z^7e^z - 2z^2 \sin(z) + \frac{z}{z^2 + 1} \). **Task**: Calculate \( g'(z) \). d. **Scenario**: A moving particle’s position, in feet, at time \( t \) seconds is given by \( s(t) = \frac{3\cos(t) - \sin(t)}{t^4} \). **Task**: Find the particle’s instantaneous velocity at \( t = 1 \). e. **Given Data**: Assume \( f(x) \) and \( g(x) \) are differentiable functions with known values: - \( f(3) = -2 \) - \( f'(3) = 7 \) - \( g(3) = 4 \) - \( g'(3) = -1 \) - **Function Definitions**: - \( p(x) = f(x) \cdot g(x) \) - \( q(x) = \frac{f(x)}{g(x)} \) **Task**: Calculate \( p'(3) \) and \( q'(3) \) using the provided values. Through these exercises, you will enhance your understanding of derivative applications and strengthen your calculus skills.
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