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Unformatted text preview: 1. [7 pts.] Find the area between y 2 cos 2x and y calculator for the simple calculations in the end.) Graphing the 2 functions, we see that the 2 points of intersection occur approximately when x 601 and when x 601. We also must subtract the bottom curve (2x2 in this case) from the top curve (2 cos 2x in this problem). Thus, we have 2 cos 2x 2x2 dx
601 24 2. [4 pts.] If t is in months and r t is in dollars per month, what are the units of 0 r t dt? Also, draw a possible sketch for the integral. (Make sure EVERYTHING is labeledaxes, function, area, ...) r(t) = $/month r(t) Here, the xaxis is in months, and the yaxis is in dollars/month. Multiplying these two together, we have dollars month dollars month 24 3. The total cost in dollars to produce q units of a product is C q . Fixed costs are $20,000. The marginal cost is C q 0 005q2 q 56 (a) [3 pts.] Find the total cost to produce 174 units. (set up by handmay use calculator to evaluate) 20 000 0 005q2 q 56 dq $23 386 04 0 (b) [3 pts.] Find C 175 using C q . Interpret your answer. (set up by handmay use calculator to evaluate) C 174 33 38 This means that it costs approximately $33.38 to make the 175th item. So the total cost, C 175 is approximately $23,386.04+33.38=$23,419.42. (You could also have taken the integral of C q from 0 to 175.) 1 Total cost Fixed costs + Variable costs
174 601 1 576 Area t (months) 2x2 . (Set up and do integral by handyou may use your 4. The figure below shows the derivative G x . If G 3
y inflection pt. G'(x) gl. min. 1 Area = 6.3 2 3 Area = 8.2 gl. max. 4 x (a) [3 pts.] Find G 0 , G 1 , and G 4 . (Show how you get each answer!) G0 G1 G3 63 82 17 Area = 1.7 inflection pt. (b) [4 pts.] Label all local maxima and minima, global maxima and minima and inflection points of G x on the graph of G x . 1 and F x 5. [6 pts.] Find an antiderivative F x if F 4 handmay use calculator at very end for simple calculations) f x sin3 x cos x dx First we need to find sin x 3 cos xdx. Let u u3 du 4. sin x. Then du C F x cos xdx So now we have Now we need to evaluate F x at x F C F x 2 Thus, 1 4 sin x 4 # $ " 4 1 sin 4 4 1 2 4 2 1 4 4 16 1 C 16 17 16 4 C C 1 1 16 16 17 16 u4 4 1 sin x 4 82 7 G 0 51 7 G1 12 G4 7 17 G 4 f x . (set up and work completely by 4 C 1 ! 7: 53 6. [3 pts.] Using the figure below, if F x is an antiderivative of f x and F 0 FTC. 25, estimate F 7 using the 0 2 4 6 8 1 2 3 4 5 6 7 x f x dx
0 7. [7 pts.] Find the exact area bounded by y 2x3 5x2 1 and the xaxis. Graph and shade this area. (Set up integral(s) by handeverything else can be done on the calculator.) y Area 2x3 5x2 1 dx 2x3 5x2 2 414 5 x 4 3752 4 979 603965 8. (a) [4 pts.] Estimate 030 f t dt using Riemann sums and the following table. (set up by handmay use calculator for calculations) Year E 1940 6.9 1950 9.4 1960 13.0 1970 18.5 1980 20.9 1990 19.6 3 5 ' ' F 7
7 2 & % 4 f x 7 F 0 F 7 f x dx 11 5 25 0 2 6 22 25 414 1 dx 293 409 2 (b) [1 pt.] What is n? (c) [1 pt.] What is t? 3 10 (d) [4 pts.] Represent the righthandsum of the above integral graphically.
E 18 14 10 6 2 t 1950 1960 1970 9. Integrate the following problems. Simplify completely with coefficients in front and positive exponents! 2
3 Let u 2 3du 3 (b) [7 pts.] Let u 2e x 1. Then du 2e x dx 5 C 4 10 0 1 1 du u 2 5 2 1 du u 5
x 2e 1 ex dx So we have 1 2 du e x dx 5 ln u 2 5 ln 2 2 ex 1 # 3 2 u1 2 u1 2 du C 2u3 2 C 22 3 x C ) )
3u3 ) x1 3. Then du 1 2 3 dx 3x 3du 2 ) # # (a) [7 pts.] ( lhs: 10(6.9+9.4+13) = 293 rhs: 10(9.4+13+18.5) = 409 Averaging the two together, we have 351 3 x x2 dx So we have ) ) ) ) x 2 3 dx. 3 2 C (c) [7 pts.] 5 5 ln x 3 x2 x3 dx dx 1 2 C 10. Consider the improper integral (a) [4 pts.] Find x 0 e dx. b x 0 e dx using the FTC. e dx
0 x 1 e b e0 0 (b) [2 pts.] Evaluate (a) when b 100 1000 10 000. What does this tell you about the improper integral? (set up by handmay use calculator to evaluate) e 100 1; e 1000 1; e 10 000 0 11. [4 pts.] List the following integrals in descending order based on the graph below. Also tell whether each integral is positive, negative, or approximately 0. f x dx pos
c f x dx pos
b f x dx pos
a f x dx neg
a y f x a b c d e x 5 47 98 46 45 e d e 3 e x dx 1 Thus 2 b e x b e b 1 1 c 2 3 3 2 x1 # ) ) ) # 1 3 1 2 1 x 3 1 x 2 1 x1 3 3 1 3 ) ) (d) [7 pts.] 10 0 5 dx x 5 dx x 1 dx x C # 1 x 2 1 2 3 1 x C 12. A bicyclist is pedaling along a straight road for one hour with a velocity v shown in the figure below. She starts out eight kilometers from the lake and positive velocities take her towards the lake. v (km/hr)
10 5 0 5 10 15 20 25 1 t (hours) (a) [3 pts.] Does the cyclist ever turn around? If so, at what time(s)? Yes, after 20 minutes. (b) [3 pts.] When is she going the fastest? How fast is she going then? Toward the lake or away? After 40 minutes, she is going away from the lake at 25 km/hr. (c) [3 pts.] When is she closest to the lake? Approximately how close to the lake does she get? After 20 minutes, she is approximately 8
5 3 1 1 6 3 km from the lake. Each rectangle is 6 x5 (d) [3 pts.] When is she farthest from the lake? Approximately how far from the lake is she then? She is furthest from the lake after 1 hour. She is approximately 8 13 6 5 3 5 6 17 17 km away. 5 6 . ...
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This note was uploaded on 04/29/2008 for the course ECON 120B taught by Professor Jeon during the Winter '08 term at UCSD.
 Winter '08
 Jeon
 Microeconomics

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