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height above the ground at the base of the cliff is H(t) = -16t2 + 40t +24 feet. a. Find the velocity of the ball after two seconds. b. When does the ball hit the ground, and what is its impact velocity? c. When does the ball have a velocity of zero? What physical interpretation should be given to this time?

height above the ground at the base of the cliff is H(t) = -16t2 + 40t +24 feet. a. Find the velocity of the ball after two seconds. b. When does the ball hit the ground, and what is its impact velocity? c. When does the ball have a velocity of zero? What physical interpretation should be given to this time?

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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Question 32

$ider the Tunction de. by 400 1 if x < 0 300 f(x) = -x +1 1 200 if x > 0 100 x +1 X 10 20 30 40 50 60 Sketch this graph and find all points where the graph is continuous but not differentiable. temperature Figure 2.53 Enzyme activity as a function of temperature. 30. Consider the function defined by 0 ifx <0 34. In Example 7 we discussed how the extent of Arctic sea ice is modeled by the function if 0 < x < 1 f(x) = 1 if x > 1 S(t) = 7.292 + 0.023t – 0.004t² million square kilometers Sketch this graph and find all points where the graph is continuous but not differentiable. where t is years after 1980. Use the definition of the derivative to compute S'(20). How does this value compare to S'(30) found in Example 7? What does this comparison suggest about the rate at which arctic ice is being lost? Level 2 APPLIED AND THEORY PROBLEMS 31. Abaseball is thrown upward and its height at time t in seconds is given by 35. In 2010, W. B. Grant published an article about the prevalence of multiple sclerosis in three U.S. communities and the role of vitamin D. The best- H(t) = 64t – 16t2 feet a. Find the velocity of the baseball after two fitting quadratic relationship to the data published in this article, relating prevalence of multiple sclerosis (MS) to latitude (exposure to sunlight, hence vita- min D synthesis decreases with latitude) is shown in Figure 2.54. This quadratic function is given by seconds. b. Find the time at which the baseball hits the ground. c. Find the velocity of the baseball when it hits the ground. Р(x) — 499 — 30.8х + 0.5х? сases per 100,000 32. A ball is thrown directly upward from the edge of a cliff and travels in such a way that t seconds later, its where x is latitude. enzyme activity
Transcribed Image Text:$ider the Tunction de. by 400 1 if x < 0 300 f(x) = -x +1 1 200 if x > 0 100 x +1 X 10 20 30 40 50 60 Sketch this graph and find all points where the graph is continuous but not differentiable. temperature Figure 2.53 Enzyme activity as a function of temperature. 30. Consider the function defined by 0 ifx <0 34. In Example 7 we discussed how the extent of Arctic sea ice is modeled by the function if 0 < x < 1 f(x) = 1 if x > 1 S(t) = 7.292 + 0.023t – 0.004t² million square kilometers Sketch this graph and find all points where the graph is continuous but not differentiable. where t is years after 1980. Use the definition of the derivative to compute S'(20). How does this value compare to S'(30) found in Example 7? What does this comparison suggest about the rate at which arctic ice is being lost? Level 2 APPLIED AND THEORY PROBLEMS 31. Abaseball is thrown upward and its height at time t in seconds is given by 35. In 2010, W. B. Grant published an article about the prevalence of multiple sclerosis in three U.S. communities and the role of vitamin D. The best- H(t) = 64t – 16t2 feet a. Find the velocity of the baseball after two fitting quadratic relationship to the data published in this article, relating prevalence of multiple sclerosis (MS) to latitude (exposure to sunlight, hence vita- min D synthesis decreases with latitude) is shown in Figure 2.54. This quadratic function is given by seconds. b. Find the time at which the baseball hits the ground. c. Find the velocity of the baseball when it hits the ground. Р(x) — 499 — 30.8х + 0.5х? сases per 100,000 32. A ball is thrown directly upward from the edge of a cliff and travels in such a way that t seconds later, its where x is latitude. enzyme activity
2.6 Derivative at a Point 181 23. 24. height above the ground at the base of the cliff is y H(t) = -16t? + 40t + 24 0.8 feet. 1 06 a. Find the velocity of the ball after two seconds. b. When does the ball hit the ground, and what is its impact velocity? 0.4 -2 -1 1 2 x -1 0.2 c. When does the ball have a velocity of zero? What physical interpretation should be given to this time? -2 E1.0 –0.5 0.5 1.0 33. If the data in Figure 2.53 represents a set of measure- ments relating enzyme activity to temperature in degrees Celsius, and the quadratic equation 25. f(x) = |x - 2| 26. f(x) = 2|x +1| if x < 1 x – 3 ifx > 1 (-2x 27. Let f(x) = A(x) = 11.8 + 19.1 x – 0.2 x² a. Sketch the graph of f. provides a good fit to this data, then find A'(50) and discuss its meaning. b. Show that f is continuous, but not differentiable, at x = 1. 28. Give an example of a function that is continuous on (-0, 0) but is not differentiable at x = 5. y %3| 500 29. Consider the function defined by 400 1 if x < 0 300 -x +1 1 | f(x) = 200 if x > 0 100 X +1 0 - 10 20 30 40 50 60 Sketch this graph and find all points where the graph is continuous but not differentiable. temperature Figure 2.53 Enzyme activity as a function of temperature. 30. Consider the function defined by 0 ifx <0 34. In Example 7 we discussed how the extent of Arctic if 0 enzyme activity
Transcribed Image Text:2.6 Derivative at a Point 181 23. 24. height above the ground at the base of the cliff is y H(t) = -16t? + 40t + 24 0.8 feet. 1 06 a. Find the velocity of the ball after two seconds. b. When does the ball hit the ground, and what is its impact velocity? 0.4 -2 -1 1 2 x -1 0.2 c. When does the ball have a velocity of zero? What physical interpretation should be given to this time? -2 E1.0 –0.5 0.5 1.0 33. If the data in Figure 2.53 represents a set of measure- ments relating enzyme activity to temperature in degrees Celsius, and the quadratic equation 25. f(x) = |x - 2| 26. f(x) = 2|x +1| if x < 1 x – 3 ifx > 1 (-2x 27. Let f(x) = A(x) = 11.8 + 19.1 x – 0.2 x² a. Sketch the graph of f. provides a good fit to this data, then find A'(50) and discuss its meaning. b. Show that f is continuous, but not differentiable, at x = 1. 28. Give an example of a function that is continuous on (-0, 0) but is not differentiable at x = 5. y %3| 500 29. Consider the function defined by 400 1 if x < 0 300 -x +1 1 | f(x) = 200 if x > 0 100 X +1 0 - 10 20 30 40 50 60 Sketch this graph and find all points where the graph is continuous but not differentiable. temperature Figure 2.53 Enzyme activity as a function of temperature. 30. Consider the function defined by 0 ifx <0 34. In Example 7 we discussed how the extent of Arctic if 0 enzyme activity
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