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Calculus
what is partial derivatives
Calculus
14: Give an example of a radical function f(x) whose domain is (3,∞) and whose range is (-∞ ,0]. You do not need to graph the function. Simply write it out.
Do not use this solution y = -√(x-3)
15: Give an example of a radical function f(x) that goes through the point (2,4) and whose domain and range are both (-∞ ,∞ ). You do not need to graph the function. Simply write it out. Do not use this solution y = ^3√(x-2) + 4
Do not use this solution y = -√(x-3)
15: Give an example of a radical function f(x) that goes through the point (2,4) and whose domain and range are both (-∞ ,∞ ). You do not need to graph the function. Simply write it out. Do not use this solution y = ^3√(x-2) + 4
Calculus
300
8: The function N(t) = ----------------------- describes the spread of a rumor among a group of people in
1 + 299e^-0.36t
an enclosed space. N represents the number of people who have heard the rumor, and t is measured in minutes since the rumor was started. Which of the following statements are true?
Check all that apply?
Initially, only one person had heard the rumor.
It will take 30 minutes for 100 people to hear the rumor.
There are 300 people in the enclosed space.
The rate at which the rumor spreads changes over time.
8: The function N(t) = ----------------------- describes the spread of a rumor among a group of people in
1 + 299e^-0.36t
an enclosed space. N represents the number of people who have heard the rumor, and t is measured in minutes since the rumor was started. Which of the following statements are true?
Check all that apply?
Initially, only one person had heard the rumor.
It will take 30 minutes for 100 people to hear the rumor.
There are 300 people in the enclosed space.
The rate at which the rumor spreads changes over time.
Calculus
64
7: One model of earth's population growth is P(t) = -----------------
(1 + 11e^-0.08t) , where t is measured in years since 1990, and P is measured in billions of people. Which of the following statements are true?
Check all that apply.
In 1990, there were 5.33 billion people.
The population of Earth will grow exponentially without bound.
The population of Earth is increasing by a steady rate of 8% per year.
The carrying capacity of Earth is 64 billion people.
7: One model of earth's population growth is P(t) = -----------------
(1 + 11e^-0.08t) , where t is measured in years since 1990, and P is measured in billions of people. Which of the following statements are true?
Check all that apply.
In 1990, there were 5.33 billion people.
The population of Earth will grow exponentially without bound.
The population of Earth is increasing by a steady rate of 8% per year.
The carrying capacity of Earth is 64 billion people.
Calculus
Air pressure may be represented as a function of height above the surface of earth as shown below.
p(h) = Poe^ -.00012h
1: In this function, P0 is air pressure at sea level, and h is measured in meters. Which of the following equations will find the height at which air pressure is 75% of the air pressure at sea level?
A: 0.75Po=Poe^-0.00012h
B: Po=0.75Poe^-0.00012h
C: h=0.75e^-0.00012
D: 0.75=h ∙ e^-0.00012
2: If you put $ 2,000 in a savings account that pays 6% interest compounded continuously, how much money will you have in your account in 4 years? Assume you make no additional deposits or withdraws.
A: $2,983.65
B: $2,542.50
C: $8,326.49
D: $3,168.15
p(h) = Poe^ -.00012h
1: In this function, P0 is air pressure at sea level, and h is measured in meters. Which of the following equations will find the height at which air pressure is 75% of the air pressure at sea level?
A: 0.75Po=Poe^-0.00012h
B: Po=0.75Poe^-0.00012h
C: h=0.75e^-0.00012
D: 0.75=h ∙ e^-0.00012
2: If you put $ 2,000 in a savings account that pays 6% interest compounded continuously, how much money will you have in your account in 4 years? Assume you make no additional deposits or withdraws.
A: $2,983.65
B: $2,542.50
C: $8,326.49
D: $3,168.15
Calculus
The original value of a car is $22,000 and it depreciates (loses value) by 15% each year. What is the value of the car after three years?
A: $74.25
B: $15,895
C: $56,595
D: $13,510.75
A ball dropped from a height of 12 feet and returns to a height that is one-half of the height from which it fell. How far will the ball have traveled when it hits the ground for the fifth time?
A: 34.5 feet
B: 1.5 feet
C: 12 feet
D: 24 feet
Which of the following are geometric sequences?
Check all that apply
10, 5, 2.5, 1.25, 0.625, 0.3125
5, 10, 15, 20, 25
1, 1, 2, 3, 8, 13,
-9, -3, -1, -1/3, -1/9, -1/27
A: $74.25
B: $15,895
C: $56,595
D: $13,510.75
A ball dropped from a height of 12 feet and returns to a height that is one-half of the height from which it fell. How far will the ball have traveled when it hits the ground for the fifth time?
A: 34.5 feet
B: 1.5 feet
C: 12 feet
D: 24 feet
Which of the following are geometric sequences?
Check all that apply
10, 5, 2.5, 1.25, 0.625, 0.3125
5, 10, 15, 20, 25
1, 1, 2, 3, 8, 13,
-9, -3, -1, -1/3, -1/9, -1/27
Calculus
7: What is the maximum number of relative extrema contained in the graph of this function?
f(x) = 3x^5 – x^3 + 4x – 2
Answer:______
8: In which direction does the left side of the graph of this function point?
f(x) = 3x^3 – x^2 + 4x – 2
Answer:_______
9: In which direction does the right side of this graph point?
f(x) = 3x^3 – x^2 + 4x – 2
Answer:_______
f(x) = 3x^5 – x^3 + 4x – 2
Answer:______
8: In which direction does the left side of the graph of this function point?
f(x) = 3x^3 – x^2 + 4x – 2
Answer:_______
9: In which direction does the right side of this graph point?
f(x) = 3x^3 – x^2 + 4x – 2
Answer:_______
Calculus
A man is walking from his front door to his car. The length of garden from his door to the road is 4 m. The car is parked 10 m up the road. Because of snow, he can only walk at 0.4 m/s through the garden, and 0.5 m/s on the road. What route should he take to minimize the time taken to get to his car?
Set it up like this: Let x be the from the end of the path to the point on the road that he aims for.
Then find expressions for each of the two parts of his journey, and use the corresponding speeds to determine the time taken for each part.
Find an expression for the total time and minimize it.
Set it up like this: Let x be the from the end of the path to the point on the road that he aims for.
Then find expressions for each of the two parts of his journey, and use the corresponding speeds to determine the time taken for each part.
Find an expression for the total time and minimize it.
Calculus
1. Suppose a patient is given a continuous intravenous infusion of glucose at a constant rate of
mg/min. Then, the rate at which the amount of glucose in the bloodstream is changing at
time (in minutes) because of this infusion is given by
A'(t)=re^-at
mg/min, where is a positive constant associated with the rate at which excess glucose is
eliminated from the bloodstream and is dependent on the patient’s metabolism rate.
a) Derive an expression for the amount of glucose in the bloodstream at time t if A(0)=0
b) If and mg/min, find the amount of glucose in the bloodstream 5
minutes after treatment began.
mg/min. Then, the rate at which the amount of glucose in the bloodstream is changing at
time (in minutes) because of this infusion is given by
A'(t)=re^-at
mg/min, where is a positive constant associated with the rate at which excess glucose is
eliminated from the bloodstream and is dependent on the patient’s metabolism rate.
a) Derive an expression for the amount of glucose in the bloodstream at time t if A(0)=0
b) If and mg/min, find the amount of glucose in the bloodstream 5
minutes after treatment began.
Calculus
Find the area of the largest rectangle with one corner at the origin, the opposite corner in the first quadrant on the graph of the parabola f(x)=48−4x^2, and sides parallel to the axes.
