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Questions and Answers
What does the statement C(200) = 100,000 represent?
What does the statement C(200) = 100,000 represent?
It represents that the total cost to manufacture 200 iPhones is $100,000.
What is the interpretation of C'(200) = 500?
What is the interpretation of C'(200) = 500?
This indicates that the cost of manufacturing one additional iPhone at the production level of 200 iPhones is $500.
How would you approximate the cost of manufacturing 203 iPhones?
How would you approximate the cost of manufacturing 203 iPhones?
The approximate cost for 203 iPhones is $101,500.
What is the limit of f(x) as x approaches 7 from the left, assuming f(x) is defined as squared function?
What is the limit of f(x) as x approaches 7 from the left, assuming f(x) is defined as squared function?
What value does lim f(x) as x approaches 7 from the right yield?
What value does lim f(x) as x approaches 7 from the right yield?
What is the overall limit of f(x) as x approaches 7?
What is the overall limit of f(x) as x approaches 7?
What is the limit of (2x + 3)/(4x^5 - 1) as x approaches infinity?
What is the limit of (2x + 3)/(4x^5 - 1) as x approaches infinity?
How do you calculate when an investment of $30,000 will grow to $90,000 at continuous compounding interest?
How do you calculate when an investment of $30,000 will grow to $90,000 at continuous compounding interest?
What is the limit of (x² + 1) / (x + 3) as x approaches -3?
What is the limit of (x² + 1) / (x + 3) as x approaches -3?
Calculate the limit of (x - 3) / (3x² + x - 12) as x approaches 3.
Calculate the limit of (x - 3) / (3x² + x - 12) as x approaches 3.
What is the limit of (√x - 8) / (x - 64) as x approaches 64?
What is the limit of (√x - 8) / (x - 64) as x approaches 64?
Using the limit definition, what is the derivative f'(-1) for f(x) = x² - 3x + 2?
Using the limit definition, what is the derivative f'(-1) for f(x) = x² - 3x + 2?
What is the equation of the tangent line to the curve f(x) = x² - 3x + 2 at the point (-1, 6) in slope-intercept form?
What is the equation of the tangent line to the curve f(x) = x² - 3x + 2 at the point (-1, 6) in slope-intercept form?
What is the left-hand limit of f(x) as x approaches -4?
What is the left-hand limit of f(x) as x approaches -4?
State whether f(x) is continuous at x = 1 based on the given limits.
State whether f(x) is continuous at x = 1 based on the given limits.
What is the value of f(6)?
What is the value of f(6)?
What is the limit of the function as x approaches -3 for the expression (x² + 1)/(x + 3)?
What is the limit of the function as x approaches -3 for the expression (x² + 1)/(x + 3)?
Using the limit method, what is the limit of (x - 3)/(3x² + x - 12) as x approaches 3?
Using the limit method, what is the limit of (x - 3)/(3x² + x - 12) as x approaches 3?
What is the derivative f'(-1) using the limit definition for the function f(x) = x² - 3x + 2?
What is the derivative f'(-1) using the limit definition for the function f(x) = x² - 3x + 2?
What is the equation of the tangent line to the graph of f(x) = x² - 3x + 2 at the point (-1, 6)?
What is the equation of the tangent line to the graph of f(x) = x² - 3x + 2 at the point (-1, 6)?
What is the limit of f(x) as x approaches -4 from the left?
What is the limit of f(x) as x approaches -4 from the left?
Is the function f continuous at x = 1? If not, state the reason.
Is the function f continuous at x = 1? If not, state the reason.
What is the limit of f(x) as x approaches 6 from the right?
What is the limit of f(x) as x approaches 6 from the right?
Explain why the interpretation of the limit as x approaches infinity of (2x + 3) / (4x⁵ - 1) equals 0. Include reasoning.
Explain why the interpretation of the limit as x approaches infinity of (2x + 3) / (4x⁵ - 1) equals 0. Include reasoning.
Calculate the precise value of t when $30,000$ is invested at an interest rate of $7.27%$ compounded continuously to reach $90,000$. Include necessary steps.
Calculate the precise value of t when $30,000$ is invested at an interest rate of $7.27%$ compounded continuously to reach $90,000$. Include necessary steps.
Describe the significance of C'(200) = 500 in the context of manufacturing costs.
Describe the significance of C'(200) = 500 in the context of manufacturing costs.
What method can you use to evaluate the limit of f(x) as x approaches 7, and what is the expected outcome?
What method can you use to evaluate the limit of f(x) as x approaches 7, and what is the expected outcome?
Given the function f(x) described, what can you infer about the continuity of f(x) at x = 7?
Given the function f(x) described, what can you infer about the continuity of f(x) at x = 7?
For the limit calculation of (4x^5 - 1) / (7x^5 + 1) as x approaches infinity, what is the simplification that occurs?
For the limit calculation of (4x^5 - 1) / (7x^5 + 1) as x approaches infinity, what is the simplification that occurs?
Using the cost function C(x), how would you estimate the cost of producing a small number of items beyond a given point?
Using the cost function C(x), how would you estimate the cost of producing a small number of items beyond a given point?
Why is showing all work important in math exams, particularly in related limits and cost calculations?
Why is showing all work important in math exams, particularly in related limits and cost calculations?
Flashcards
Interpreting C(200) = 100,000
Interpreting C(200) = 100,000
The total cost to manufacture 200 iPhones is $100,000.
Interpreting C'(200) = 500
Interpreting C'(200) = 500
The marginal cost of manufacturing an iPhone when producing 200 iPhones is $500.
Approximating cost to make 203 iPhones (given C(200)=100,000 and C'(200)=500)
Approximating cost to make 203 iPhones (given C(200)=100,000 and C'(200)=500)
The approximate cost to make 203 iPhones is $101,500.
Limit of f(x) as x approaches 7 from the left (lim x→7− f(x))
Limit of f(x) as x approaches 7 from the left (lim x→7− f(x))
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Limit of f(x) as x approaches 7 from the right (lim x→7+ f(x))
Limit of f(x) as x approaches 7 from the right (lim x→7+ f(x))
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Limit of f(x) as x approaches 7 (lim x→7 f(x))
Limit of f(x) as x approaches 7 (lim x→7 f(x))
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Limit of (2x + 3) / (4x⁵ - 1) as x approaches ∞
Limit of (2x + 3) / (4x⁵ - 1) as x approaches ∞
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Limit of (4x⁵ - 1) / (7x⁵ + 1) as x approaches ∞
Limit of (4x⁵ - 1) / (7x⁵ + 1) as x approaches ∞
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Evaluating limits
Evaluating limits
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L'Hopital's Rule
L'Hopital's Rule
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Limit notation
Limit notation
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Tangent line slope
Tangent line slope
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f'(x)
f'(x)
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Tangent line equation
Tangent line equation
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One-sided limit
One-sided limit
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Continuity at a point
Continuity at a point
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Marginal cost
Marginal cost
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Approximating cost using marginal cost
Approximating cost using marginal cost
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Left-hand limit
Left-hand limit
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Right-hand limit
Right-hand limit
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Limit of a function
Limit of a function
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Limit at infinity
Limit at infinity
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Continuous compounding
Continuous compounding
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Solving for time in continuous compounding
Solving for time in continuous compounding
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Limit Calculation
Limit Calculation
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Undefined Limit
Undefined Limit
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Study Notes
Exam Questions and Solutions
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Question 1a: Interpreting C(200) = 100,000. It will cost $100,000 to manufacture 200 iPhones.
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Question 1b: Interpreting C'(200) = 500. It costs $500 to make one additional iPhone (at a production level of 200 iPhones).
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Question 1c: Approximating the cost of making 203 iPhones. Calculate the cost for 200 iPhones ($100,000) plus the additional cost of making three iPhones ($500/phone * 3 phones = $1,500). Therefore, the total approximate cost is $101,500.
Limits and Continuity
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Question 2a: Finding the limit of f(x) as x approaches 7. The limit from the left (x→7−) is 48; from the right (x→7+) is 48. Thus, the limit is 48.
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Question 2b: Finding the limit of (2x + 3)/(4x⁵ − 1) as x approaches infinity. The limit is 0.
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Question 2c: Finding the limit of (4x⁵ − x)/(7x⁵ + 5x) as x approaches infinity. The limit is 4/7.
Compound Interest
- Question 3: Calculating the time to reach $90,000 in an account with $30,000 invested at 7.27% compounded continuously. The exact time is ln(3) / ln(1.0727) years.
Limits using Graph
- Question 6: Determining various limits and function values by inspecting the provided graph. Includes limits from the left and right, function values, and continuity at specified points on the graph. (Answers for each limit/function value are provided in the table; no further explanation needed)
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Description
Test your understanding of key concepts in calculus and finance with this quiz! Questions cover manufacturing costs, limits, continuity, and compound interest calculations. Assess your grasp on critical mathematical principles and their applications.
