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Course Calendar
Week 0
9/23: Section 5.1 (How do we compute distance?)
Week 1
9/26: Section 5.2 (The Definite Integral)
9/28: Section 5.3 (The First Fundamental Theorem of Calculus)
9/30: Section 5.4 (Theorems About Integrals, Part 1)
Week 2
10/3: Se
(9/1/08)
Math 10B. Lecture Examples. Section 9.2. Geometric series
Example 1 Suppose you want to go from a point A toward a second point B two miles away. First you go one mile (Figure 1). Then you go a half mile further for a total of 1 + 1 miles 2 (Figu
(9/1/08)
Math 10B. Lecture Examples. Section 9.1. Sequences
Example 1 A piece of meat at 30 C is put in a freezer at time n = 0. The temperature of the 30 (Figure 1). freezer is 0 C, and the temperature of the meat n hours later is T = n+1 30 Does the seq
(9/30/08)
Math 10B. Lecture Examples. Section 8.5. Applications to physics
Example 1 A man sliding a box exerts a force of 20 3 s pounds when the box is at s (feet) on an s-axis. How much work does he do when he moves the box from s = 0 to s = 9?
Answer:
(9/30/08)
Math 10B. Lecture Examples. Section 8.4. Density and centers of mass
Example 1 The density of a two-foot-long rod is = 2 + 6x (pounds per foot) at a distance x feet from one end. How much does it weigh?
Answer: [Weight] = 16 pounds
Example 2
Fin
(9/30/08)
Math 10B. Lecture Examples. Section 8.3. Area and arc length in polar coordinates
Example 1 Sketch the curve in an xy -plane with the polar equation r = 1 + cos , 0 2. (The curve is called a cardioid because of its heart-like shape.)
Answer: Fig
(9/30/08)
Math 10B. Lecture Examples. Section 8.2. Applications to geometry
Example 1 The region bounded by the curve y = x x4 and the x-axis is rotated about the x-axis. Find the volume of the solid that is generated. 1 Answer: Figures A1a and Figure A1b
(9/30/08)
Math 10B. Lecture Examples. Section 8.1. Areas and volumes
Example 1 The base of a solid is the region between the curves y = 1 x2 and y = 1 2 for 0 x 1 in an xy -plane and its cross sections perpendicular to the x-axis are squares. Find its vol
(9/30/08)
Math 10B. Lecture Examples. Section 7.7. Improper integrals
Example 1
Evaluate
3
1 dx. x3
1 18
Answer:
3
1
x3
dx =
Example 2
Find the area of the region between y = 1/x and the x-axis for x 1. Answer: Figure A1 [Area] = y y= 1
1 x
Figure A1
1 Is
(9/30/08)
Math 10B. Lecture Examples. Section 7.5. Approximating denite integrals
1
Example 1
Calculate the Midpoint Rule approximation of
0
x2 dx corresponding to
the partition of [0,1] into ve equal subintervals. Draw the curve y = x2 with the rectangle
(9/30/08)
Math 10B. Lecture Examples. Section 7.4. Algebraic identities and trigonometric substitutions
Example 1 Find the partial-fraction decomposition of result a common denominator.
Answer:
1 . Check by giving the x(x 1)
1
x(x 1)
=
1
x1
1
x
Check:
1
x
(9/7/08)
Math 10B. Lecture Examples. Section 9.3. Convergence of series
Example 1
Does
n=2
1 n(ln n)2
1 =
converge or diverge?
1 The improper integral and the innite series converge. (The rst 24 partial ln(2) 2 sums of the series are plotted in Figure A1.
Math 10B Test 1 Solutions
100 pts
October 21, 2009
Professor Evans
Directions: Answers alone are not sucient. Show all work. Each problem is 20 points. (1) Let f (x) denote the function ex . Recall that e = 2.71828 . . . . Jane partitioned the interval [0
Math 10B Test 2
100 pts
November 23, 2009
Professor Evans
Directions: Answers alone are not sucient. Show all work. Each problem is worth 20 points. (1) On January 2, 2008, Jack was given a $100 savings account and Jane was given a $60 savings account. On
Winter 2014
Math 10B
Midterm Exam 1 vA
University of California, San Diego
Department of Mathematics
Instructions
1. Write your Name, PID, Section, and Exam Version on the front of your Blue Book.
2. No calculators or other electronic devices are allowed
(9/1/08)
Math 10B. Lecture Examples. Section 11.5. Growth and Decay
Example 1 Match problems (I) through (IV) below to dierential equations (a) through (d) and to the slope elds in Figures 1 through 4.
(I) The thickness of the ice on a lake grows at a rat
(9/8/08)
Math 10B. Lecture Examples. Section 11.4. Separation of variables
Example 1 Figure 1 shows the slope eld of the dierential equation dy =y dx and Figure 2 shows the graphs of eight solutions. (a) Use the dierential equation to explain the pattern
(9/1/08)
Math 10B. Lecture Examples. Section 11.3. Eulers method
Example 1 Figure 1 shows the slope eld for the dierential equation, dy = (1 x)y. dx Draw the graph of approximate solution y = yE (x) for 0 x 4 with the initial value y (0) = 1 that is obtai
(9/1/08)
Math 10B. Lecture Examples. Section 11.2. Slope elds
Example 1 dy = 1 (x y ) at the twenty 2 dx points with coordinates x = 0, 1, 2, 3, 4 and y = 0, 1, 2, 3 in Figure 1. (b) Describe the patterns of the slope lines and explain how they are determ
(9/1/08)
Math 10B. Lecture Examples. Section 10.3. Finding and using Taylor series
Example 1
Use the Taylor series sin x =
n=0
(1)n 2n+1 x for y = sin x centered at x = 0 to (2n + 1)!
give the Taylor series centered at x = 0 for y = sin(2x).
Answer: sin(2
(9/1/08)
Math 10B. Lecture Examples. Section 10.1. Taylor polynomials
Example 1 Find the rst-, second-, and third-degree Taylor polynomial approximations of y = ln x, centered at x = 1. 1 1 Answer: P1 (x) = x 1 P2 (x) = (x 1) 1 (x 1)2 P3 (x) = (x 1) 2 (x
(9/1/08)
Math 10B. Lecture Examples. Section 9.5. Power series and intervals of convergence
Example 1 Find the radius of convergece of the power series
j =1
(1)j +1 j 1 x = x 2 x2 + 1 x3 1 x4 + . 3 4 j
Answer: [Radius of convergence] = 1 (Figure A1a shows
(9/1/08)
Math 10B. Lecture Examples. Section 9.4. Tests for convergence
Example 1
Does the series
n=0
(0.6)n converge? n+1
Answer:
n=0
(0.6)
n
n+1
converges by the Comparison Test with the convergent Geometric Series
n=0
(0.6)n .
(All but the rst partial
Math 10B Final Exam Solutions
150 pts
December 8, 2009
Professor Evans
Directions: Answers alone are not sucient. Justify, and show all work. The multiple part problems 4,6,7 are worth 18 points each. The other six problems are worth 16 points each. (1) A
(9/30/08)
Math 10B. Lecture Examples. Section 7.3. Tables of integrals
Example 1 Find the area of the region between y =
1 and the x-axis for x(3x + 6)
1 x 3. Use the following formula from a table of integrals:
1
(ax + b)(cx + d)
dx =
1 ax + b ln + C for
(9/30/08)
Math 10B. Lecture Examples. Section 7.2. Integration by parts
Example 1 (a) Find the antiderivative dierentiation.
Answer: (a)
x cos x dx. (b) Check the result by
d (x sin x + cos x) = x cos x. dx
x cos x dx = x sin x + cos x + C (b) Product Rul
Name: Sec. No: Math 10B Midterm 2 August 26, 2010
PID: Sec. Time:
Please turn o and but away all electronic devices, except for calculators. You may use 1 page of handwritten notes, but no other resources during the exam. Show all of your work, no credit
Name: Sec. No: Math 10B Midterm 1 August 12, 2010
PID: Sec. Time:
Please turn o and but away all electronic devices, except for calculators. You may use 1 page of handwritten notes, but no other resources during the exam. Show all of your work, no credit
Name: Sec. No: Math 10B Midterm 1 August 12, 2010
PID: Sec. Time:
Please turn o and but away all electronic devices, except for calculators. You may use 1 page of handwritten notes, but no other resources during the exam. Show all of your work, no credit