Docsity
Docsity
Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity

Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips
Guidelines and tips
Sell on Docsity
Sell on Docsity
Docsity AI
Docsity AI
Log inSign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search for your university

Find the specific documents for your university's exams

Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents
download point icon20 Points

For each uploaded document

Answer questions
download point icon5 Points

For each given answer (max 1 per day)

All the ways to get free points
Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

9 Problems on Calculus - Second Midterm Examination - Fall 2004 | MATH 1A, Exams of Calculus

University of California - BerkeleyCalculus
Prof. M. Haiman

Material Type: Exam; Professor: Haiman; Class: Calculus; Subject: Mathematics; University: University of California - Berkeley; Term: Fall 2004;

Typology: Exams

2010/2011

Uploaded on 05/11/2011

koofers-user-1g8
koofers-user-1g8 🇺🇸

10 documents

1 / 2

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
Math 1A, Calculus Second Midterm Exam Haiman, Fall 2004
Name Student ID Number
Section
You may use one sheet of notes. No other notes, books or calculators. There are 9
questions, on front and back. Write answers on the exam and turn in only this paper. Show
enough work so that we can see how you arrived at your answers.
1. (10 pts) Find d2
dx2(sec x).
2. (12 pts) Differentiate x(ex).
3. (10 pts) If h(x) = f(g(x)) and f(0) = 0, g(0) = 1, f0(0) = 2, g0(0) = 3, f0(1) = 4,
g0(1) = 5, find h0(0).
4. (12 pts) If x2+y3= 17 and dx/dt = 10, find dy/dt when x= 3.
5. (11 pts) A cube is measured to be 6cm on each side, with a possible error of ±.5cm.
Use a linear approximation or differentials to estimate the error in computing the volume of
the cube.
1
pf2

Partial preview of the text

Download 9 Problems on Calculus - Second Midterm Examination - Fall 2004 | MATH 1A and more Exams Calculus in PDF only on Docsity!

Math 1A, Calculus Second Midterm Exam Haiman, Fall 2004

Name Student ID Number

Section

You may use one sheet of notes. No other notes, books or calculators. There are 9 questions, on front and back. Write answers on the exam and turn in only this paper. Show enough work so that we can see how you arrived at your answers.

  1. (10 pts) Find d

2 dx^2 (sec^ x).

  1. (12 pts) Differentiate x(e x) .
  2. (10 pts) If h(x) = f (g(x)) and f (0) = 0, g(0) = 1, f ′(0) = 2, g′(0) = 3, f ′(1) = 4, g′(1) = 5, find h′(0).
  3. (12 pts) If x^2 + y^3 = 17 and dx/dt = 10, find dy/dt when x = 3.
  4. (11 pts) A cube is measured to be 6cm on each side, with a possible error of ±.5cm. Use a linear approximation or differentials to estimate the error in computing the volume of the cube.
  1. (12 pts) Find all local and absolute minima and maxima of the function f (x) = x^2 (x + 6) on the interval [− 5 , 3].
  2. (11 pts) Verify that f (x) = x^3 + x − 1 satisfies the hypotheses of the Mean Value Theorem on the interval [0, 2], and find all points c for which the conclusion of the Mean Value Theorem holds.
  3. (10 pts) Compute lim x→ 0

x + x^2 ex^ − e−x

  1. (12 pts) Use the information below to sketch the graph of y = (x − 1)/x^2. Show any local or absolute maxima and minima and any inflection points by plotting them on your sketch and labelling them with their x and y coordinates.
  • The domain of f (x) = (x − 1)/x^2 is (−∞, 0) ∪ (0, ∞).
  • limx→ 0 + (x − 1)/x^2 = limx→ 0 − (x − 1)/x^2 = −∞.
  • limx→∞(x − 1)/x^2 = limx→−∞(x − 1)/x^2 = 0.
  • y = 0 at x = 1, y < 0 on (−∞, 0) ∪ (0, 1), and y > 0 on (1, ∞).
  • y′^ = (2 − x)/x^3 ; y′^ = 0 at x = 2, y′^ < 0 on (−∞, 0) ∪ (2, ∞), and y′^ > 0 on (0, 2).
  • y′′^ = (2x − 6)/x^4 ; y′′^ = 0 at x = 3, y′′^ < 0 on (−∞, 0) ∪ (0, 3), and y′^ > 0 on (3, ∞).