Exam3solus(axelson)

Exam3solus(axelson) - 1. [7 pts.] Find the area between y 2...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 x-axis is in months, and the y-axis 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 x-axis. 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 right-hand-sum 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.] ( l-h-s: 10(6.9+9.4+13) = 293 r-h-s: 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 . ...
View Full Document

This note was uploaded on 04/29/2008 for the course ECON 120B taught by Professor Jeon during the Winter '08 term at UCSD.

Ask a homework question - tutors are online