Podcast
Questions and Answers
What is the derivative of the function $f(x) = 2x ext{ln}(x)$?
What is the derivative of the function $f(x) = 2x ext{ln}(x)$?
$f'(x) = 2 ext{ln}(x) + 2$
Calculate the limit: $\lim_{x \to 0} \frac{\sin(x)}{x^4}$.
Calculate the limit: $\lim_{x \to 0} \frac{\sin(x)}{x^4}$.
$0$
What is the limit of $\lim_{x \to 0^+} \frac{\cos(x)}{x}$?
What is the limit of $\lim_{x \to 0^+} \frac{\cos(x)}{x}$?
$\infty$
What is the derivative of $f(x) = \ln |x \sec(x) + 1|$?
What is the derivative of $f(x) = \ln |x \sec(x) + 1|$?
Explain how to approach the integral for $\int 2x \ln(x) , dx$.
Explain how to approach the integral for $\int 2x \ln(x) , dx$.
Determine the limit: $\lim_{x \to 0} \frac{\sqrt{x}}{x}$.
Determine the limit: $\lim_{x \to 0} \frac{\sqrt{x}}{x}$.
What type of functions are being derived and integrated in the exam?
What type of functions are being derived and integrated in the exam?
What is a critical aspect of the exam regarding justifying answers?
What is a critical aspect of the exam regarding justifying answers?
What is the first step to solve the integral ( \int (3x - 1)^2 , dx )?
What is the first step to solve the integral ( \int (3x - 1)^2 , dx )?
Describe how to find the local extrema of the function ( f(x) = 5x^{2/3} + x^{5/3} ).
Describe how to find the local extrema of the function ( f(x) = 5x^{2/3} + x^{5/3} ).
How do you evaluate the tangent line at ( x = 1 ) for the curve ( y = \int_{0}^{3x} \cos(\pi t) , dt + 2x )?
How do you evaluate the tangent line at ( x = 1 ) for the curve ( y = \int_{0}^{3x} \cos(\pi t) , dt + 2x )?
What is the total area enclosed by the curves ( y = e^x ) and ( y = e^{2-x} ) from ( x = 0 ) to ( x = 2 )?
What is the total area enclosed by the curves ( y = e^x ) and ( y = e^{2-x} ) from ( x = 0 ) to ( x = 2 )?
What are the dimensions of an open-topped box with a square base and volume 32 that minimizes material usage?
What are the dimensions of an open-topped box with a square base and volume 32 that minimizes material usage?
How is the maximum distance traveled by a train that accelerates and decelerates at ( 4 , m/s^2 ) determined?
How is the maximum distance traveled by a train that accelerates and decelerates at ( 4 , m/s^2 ) determined?
What is the significance of the limit in calculus?
What is the significance of the limit in calculus?
How do you find inflection points of a function?
How do you find inflection points of a function?
Flashcards
Derivative
Derivative
The process of finding the rate of change of a function at a specific point.
Limit
Limit
A limit that approaches a specific value as the input approaches a certain value.
Integral
Integral
Finding the total area under a curve between two points.
Derivatives of Natural Log and Exponential Functions
Derivatives of Natural Log and Exponential Functions
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Differentiation
Differentiation
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Integration
Integration
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Composite Functions
Composite Functions
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Riemann Sum
Riemann Sum
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Finding Extrema and Inflection Points
Finding Extrema and Inflection Points
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Tangent Line
Tangent Line
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Area Between Curves
Area Between Curves
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Optimization Problems
Optimization Problems
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Acceleration
Acceleration
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Definite Integral
Definite Integral
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Antiderivative
Antiderivative
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Study Notes
MATH 1A FINAL (PRACTICE 1)
- Instructions:
- Do not turn over until instructed
- Write name and SID on each page
- Exam has 10 questions
- Exam duration: 3 hours
- Do not remove pages
- Extra scratch paper on back, label clearly
- Calculators are not permitted
- Show all work, partial credit possible
- No justification, no credit
Exam Content Overview
- Question 1: Calculate derivatives of functions (2x ln(x)/sin(x) and ln|xsec(x)+1|) without limit definition.
- Question 2: Calculate limits (lim x→0- sin(x)/x4 and lim x→0+ cos(x)).
- Question 3: Calculate integrals (integral (√x - 1)2dx from 3 to 4 and integral of x ln(x) / (2x) dx from 1 to e3).
- Question 4: Determine local extrema and inflection points of the function f(x) = 5x(2/3) + x(5/3).
- Question 5: Find the equation of the tangent line at x=1 for y = integral of cos(Ï€t)/2x dt from 0 to 3x.
- Question 6: Calculate the total area of the region enclosed by y = ex and y = e2-x between x=0 and x=2.
- Question 7: Design an open-topped box with a square base and volume 32 to minimize material amount.
- Question 8: Determine maximum distance a train can travel in 1 minute if accelerating/decelerating at 4 meters per sec2 and maximum speed is 60 meters per second.
- Question 9: Calculate the limit lim n→∞ Σni=1 (2i/ n3 + i3).
- Question 10: Find the volume of a solid with base given by ellipse x2/4 + y2/9 = 1 and equilateral triangular cross-sections parallel to the y-axis. (Use area of equilateral triangle with side length l, 13l2/4.)
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