MATH 1A FINAL (PRACTICE 1)

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Questions and Answers

What is the derivative of the function $f(x) = 2x ext{ln}(x)$?

$f'(x) = 2 ext{ln}(x) + 2$

Calculate the limit: $\lim_{x \to 0} \frac{\sin(x)}{x^4}$.

$0$

What is the limit of $\lim_{x \to 0^+} \frac{\cos(x)}{x}$?

$\infty$

What is the derivative of $f(x) = \ln |x \sec(x) + 1|$?

$f'(x) = \frac{\sec(x) + x\sec(x)\tan(x)}{x \sec(x) + 1}$ Signup and view all the answers

Explain how to approach the integral for $\int 2x \ln(x) , dx$.

Use integration by parts, letting $u = \ln(x)$ and $dv = 2x , dx$. Signup and view all the answers

Determine the limit: $\lim_{x \to 0} \frac{\sqrt{x}}{x}$.

$\infty$ Signup and view all the answers

What type of functions are being derived and integrated in the exam?

Elementary functions including logarithmic, trigonometric, and polynomial functions. Signup and view all the answers

What is a critical aspect of the exam regarding justifying answers?

Justifying your answers is essential for partial credit, even if the final answer is incorrect. Signup and view all the answers

What is the first step to solve the integral ( \int (3x - 1)^2 , dx )?

Expand ( (3x - 1)^2 ) to get ( 9x^2 - 6x + 1 ). Signup and view all the answers

Describe how to find the local extrema of the function ( f(x) = 5x^{2/3} + x^{5/3} ).

Take the derivative, set it to zero, and solve for ( x ). Signup and view all the answers

How do you evaluate the tangent line at ( x = 1 ) for the curve ( y = \int_{0}^{3x} \cos(\pi t) , dt + 2x )?

Calculate ( y ) at ( x = 1 ) and find the derivative to get the slope, then use the point-slope form. Signup and view all the answers

What is the total area enclosed by the curves ( y = e^x ) and ( y = e^{2-x} ) from ( x = 0 ) to ( x = 2 )?

Calculate ( \int_0^2 (e^{2-x} - e^x) , dx ). Signup and view all the answers

What are the dimensions of an open-topped box with a square base and volume 32 that minimizes material usage?

The base side length should be ( 4 ) units and height ( 2 ) units. Signup and view all the answers

How is the maximum distance traveled by a train that accelerates and decelerates at ( 4 , m/s^2 ) determined?

Calculate the distance during acceleration to max speed and then during deceleration. Signup and view all the answers

What is the significance of the limit in calculus?

Limits help understand the behavior of functions as they approach specific points. Signup and view all the answers

How do you find inflection points of a function?

Compute the second derivative and identify where it changes sign. Signup and view all the answers

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(πt)/2x dt from 0 to 3x.
Question 6: Calculate the total area of the region enclosed by y = e^x and y = e^2-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→∞ Σⁿ_i=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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Description

This practice quiz is designed to help students prepare for the MATH 1A final exam. It includes questions on derivatives, limits, integrals, and functions analysis. Test your knowledge and readiness to tackle calculus concepts effectively.