MATH 20B MIDTERM #2 REVIEW FORMAT OF MIDTERM #2 The format will be the same as the practice midterms. There will be six main questions worth 0 points each. These questions will be similar to problems you have done on your homework. There will be one bonus question worth 5 extra credit points. I advise you do the bonus question only if you have finished and feel good about the six main questions on the exam. There will be a survey question worth 5 extra credit points. EXAM POLICY You may bring one 8.5 by inch sheet of notes, which may be written on both sides, to each exam. You are to turn off your phones and other electronic devices during the exam. No calculators will be allowed during exams. SHOWING YOUR WORK AND PARTIAL CREDIT The same policies as Midterm # hold. You will be required to show a reasonable amount of work on each and every problem.
You will of course be deducted points for calculation errors, but you will still get most of the possible points on a problem if your steps are, for the most part, correct. You are not required to do arithmetic to simplify your answer. You do not have to evaluate logarithms (for example, you may leave ln 2 as is) or powers of e (for example, you may leave e as is). Please do, however, evaluate trigonometric functions where possible (for example don t leave sin π as is, write 0 instead). * It may be helpful to brush up on the unit circle. WHAT'S GOING TO BE ON THE EXAM? The exam is cumulative with an emphasis on Sections 7.6, 7.8,.3,.4, and Supplements - 5. MIDTERM # The material from Midterm # won t be touched on much at all. However, anything that we have covered in class is fair game. REVIEW OF SELECTED TOPICS WARNING: The following is not a comprehensive list of things to know for the exam. POLAR COORDINATES Here are the conversion formulas between rectangular and polar coordinates: Another formula which is useful: x = r cos θ and y = r sin θ. r = x 2 + y 2 = r 2 = x 2 + y 2. Area bounded by a polar curve r = f(θ): area under the curve = 2 2 β α f(θ) 2 dθ.
Area between two polar curves r = f (θ) and r = f 2 (θ): area between curves = 2 β DOING POLAR AREA PROBLEMS α (f 2 (θ) 2 f (θ) 2 ) dθ. Here are some pointers on doing polar area problems: Try to sketch the graph of what you are trying to find the area of. On some problems, the graph will be given to you In general, a good strategy for drawing a graph is to plot a few points (you can make a table of θ vs. r values). Identify what the curves bounding your region are. Figure out what your bounds are. If your boundary is made of up multiple pieces, figure out to what θ each endpoint corresponds. You can do this by calculating intersection points, or, if the endpoint is at the origin, by solving f(θ) = 0. Be careful of cases where r is negative. The θ corresponding to a negative value of r as it appears on the graph is actually θ + π. Your lower bound is always a smaller number than your upperbound and the two bounds are within 2π of each other. The lower bound should be the extreme point of the curve in the clockwise direction. To adjust your bounds, add or subtract multiples of 2π. EXAMPLE. Find the region of overlap between the circles r = sin θ and r = cos θ. COMPLEX NUMBERS Here are a few selected items. COMPLEX EXPONENTIALS e ix = cos x + i sin x cos(x) = eix + e ix 2 sin(x) = eix e ix 2i 3
MULTIPLYING AND DIVIDING COMPLEX EXPONENTIALS e a e b = e (a+b) e a and e = b e(a b). POWERS OF COMPLEX EXPONENTIALS ( re ix ) n = r n e inx = r n (cos nx + i sin nx). INTEGRATION OF COMPLEX EXPONENTIALS For any complex number α 0, e αx dx = α eαx + C. EXAMPLE. Evaluate e x cos x dx. SIMPLIFYING YOUR ANSWER Take any a + bi and multiply by a bi a bi to get a + bi = a + bi = a bi a 2 + b 2. PARTIAL FRACTION EXPANSION Review the previous lecture notes for all of the different cases and techniques for computing the partial fraction expansion. POLYNOMIAL LONG DIVISION Some of you have read about this in the textbook. If we have a non-proper rational function a 0 + a x + + a m x m b 0 + b x + + b n x n, (non-proper means m n), then we can reduce the problem using polynomial long division. EXAMPLE. Simplify x 3 + x 2 2x + 5 with polynomial long division. 4
EXAMPLE. Simplify x 2 x 2 with polynomial long division. For more information on polynomial long division, Wikipedia has an article on it. IMPROPER FRACTIONS We talked about this topic the past couple lectures. COMPARISON THEOREM Look at the dominant term of the integral to figure out whether or not it converges or diverges. Then follow the Comparison Theorem to show that the interval converges/diverges. What is the dominant term? If the integral is towards ±, then x a dominates x b if a > b. An exponential e cx dominates powers of x. The natural log ln(x) is dominated by powers of x. The trig functions sin x and cos x stay between and and hence are dominated by anything that goes to. If the integral goes to 0 and the integrand is discontinuous, then x b dominates x a if a > b (the opposite of the case). We also see that ln(x) is dominated by if c > 0. xc EXAMPLE. Does EXAMPLE. Does 0 NUMERICAL INTEGRATION Remember the formulas: TRAPEZOID APPROXIMATION The N th trapezoidal approximation to dx converge or diverge? x + 2x 2 + 3x 3 + 4x4 dx converge or diverge? x + 2x 2 + 3x 3 + 4x4 b a f(x) dx is T N = x (f(a) + 2f(a + x) + + 2f(a + (N ) x) + f(b)) 2 5
where x = b a N EXAMPLE. Calculate T 4 of 3 MIDPOINT APPROXIMATION The N th midpoint approximation to 0 sin(x) dx. b a f(x) dx is M N = x (f(a + [/2] x) + f(a + [3/2] x) + + f(a + [N /2] x)) where x = b a N EXAMPLE. Calculate M 4 of 3 0 sin(x) dx. OVER AND UNDERESTIMATING The concavity tells whether T N or M N is an over or underestimate. T N Concave Up f (x) > 0 Overestimate Concave Down f (x) < 0 Underestimate M N Concave Up f (x) > 0 Underestimate Concave Down f (x) < 0 Overestimate 6