et g(x) be a function for which g'(x)=ex (9-x²). (a) Construct a first derivative sign chart for g. Clearly state the critical numbers of g and any locations where g has a relative or global maximum or minimum, with justification. (b) Find g" (x) and then factor g"(x) to determine where g"(x)= 0 (note: you will need to use the quadratic formula to find the values where g"(x) = 0). (c) Construct a second derivative sign chart for g. Where does g have inflection points?
et g(x) be a function for which g'(x)=ex (9-x²). (a) Construct a first derivative sign chart for g. Clearly state the critical numbers of g and any locations where g has a relative or global maximum or minimum, with justification. (b) Find g" (x) and then factor g"(x) to determine where g"(x)= 0 (note: you will need to use the quadratic formula to find the values where g"(x) = 0). (c) Construct a second derivative sign chart for g. Where does g have inflection points?
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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Please give me answer very fast in 5 min sini
![et g(x) be a function for which g'(x) = e(9-x²).
(a) Construct a first derivative sign chart for g. Clearly state the critical numbers of g and any locations where g has
a relative or global maximum or minimum, with justification.
(b) Find g(x) and then factor g" (x) to determine where g"(x) = 0 (note: you will need to use the quadratic formula
to find the values where g(x) = 0).
(c) Construct a second derivative sign chart for g. Where does g have inflection points?
(d) Later this semester, we will learn why g'(x)0 as xoo: that is, we'll be able to show that g'(x) approaches
zero as x gets bigger and bigger without bound. For now, accept and use this fact together with your work in
(a)-(c) to construct a possible graph of y= g(x). (Think about what it means for the graph of g(x) to have g'(x)
to approach 0 as x gets bigger and bigger.)
Consider the one-parameter family of functions given by f(x) = x² + sin(kx), where k is a positive constant. (This
problem looks long, but most of the questions are graphical and straightforward; don't be intimidated by it :-))
Use Desmos to plot f(x) = x² +sin (kx), making a slider for k with values that range from k = 0.1 to k = 10. Experiment
with the slider to see how k affects f.
(a) By experimenting with the slider, find a value of k for which f(x) appears to have exactly two critical numbers.
Show your graph (include a screenshot, or sketch by hand) and write a sentence to explain your thinking, plus
state/label the critical numbers.
(b) By experimenting with the slider, find a value of k for which f(x) appears to have exactly four critical numbers.
Show your graph (include a screenshot, or sketch by hand) and write a sentence to explain your thinking, plus
state/label the critical numbers.
(c) Find formulas for both f'(x) and f"(x). Each will involve k.
(d) By experimenting with the slider, what is the smallest value of k for which f visually appears to have at least one
inflection point?
(e) Use your formula for f"(x) to explain why if k> √2, f(x) has infinitely many inflection points.
Let g(x)=x³-118x² +3757x-29820. Use calculus to find the absolute minimum and absolute minimum of g on the
interval [7,75]. Your work must include (a) the exact critical numbers of g (not as decimals), (b) decimal approximations
of the critical numbers accurate to 6 decimal places, and (c) appropriate function values that are accurate to 6 decimal
places, plus (d) a one-sentence summary that clearly states your conclusions.
How do your results change (if at all) if we instead consider the interval [5,73]?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0143c836-8f2c-4043-b7b3-58e093f2fd4b%2Fc11fafd6-508c-4d09-a690-750b1397994b%2Fvo6g3nh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:et g(x) be a function for which g'(x) = e(9-x²).
(a) Construct a first derivative sign chart for g. Clearly state the critical numbers of g and any locations where g has
a relative or global maximum or minimum, with justification.
(b) Find g(x) and then factor g" (x) to determine where g"(x) = 0 (note: you will need to use the quadratic formula
to find the values where g(x) = 0).
(c) Construct a second derivative sign chart for g. Where does g have inflection points?
(d) Later this semester, we will learn why g'(x)0 as xoo: that is, we'll be able to show that g'(x) approaches
zero as x gets bigger and bigger without bound. For now, accept and use this fact together with your work in
(a)-(c) to construct a possible graph of y= g(x). (Think about what it means for the graph of g(x) to have g'(x)
to approach 0 as x gets bigger and bigger.)
Consider the one-parameter family of functions given by f(x) = x² + sin(kx), where k is a positive constant. (This
problem looks long, but most of the questions are graphical and straightforward; don't be intimidated by it :-))
Use Desmos to plot f(x) = x² +sin (kx), making a slider for k with values that range from k = 0.1 to k = 10. Experiment
with the slider to see how k affects f.
(a) By experimenting with the slider, find a value of k for which f(x) appears to have exactly two critical numbers.
Show your graph (include a screenshot, or sketch by hand) and write a sentence to explain your thinking, plus
state/label the critical numbers.
(b) By experimenting with the slider, find a value of k for which f(x) appears to have exactly four critical numbers.
Show your graph (include a screenshot, or sketch by hand) and write a sentence to explain your thinking, plus
state/label the critical numbers.
(c) Find formulas for both f'(x) and f"(x). Each will involve k.
(d) By experimenting with the slider, what is the smallest value of k for which f visually appears to have at least one
inflection point?
(e) Use your formula for f"(x) to explain why if k> √2, f(x) has infinitely many inflection points.
Let g(x)=x³-118x² +3757x-29820. Use calculus to find the absolute minimum and absolute minimum of g on the
interval [7,75]. Your work must include (a) the exact critical numbers of g (not as decimals), (b) decimal approximations
of the critical numbers accurate to 6 decimal places, and (c) appropriate function values that are accurate to 6 decimal
places, plus (d) a one-sentence summary that clearly states your conclusions.
How do your results change (if at all) if we instead consider the interval [5,73]?
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In part (d), how would a constructed graph look for g(x)?
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