supporting work will receive a score of zero.essay

Math 131 Fall 2021 Examination 1 Title Be certain to point out all work to obtain full credit score for every query. Solutions with out supporting work will obtain a rating of zero. Additionally, please field your ultimate solutions, do your greatest to jot down clearly and legibly, and write your identify on each odd-numbered web page. clear Predict A number of Alternative / Brief Reply 1. (1 pt every) Beneath are four graphs, every illustrating a Riemann Sum. (a) (b) (c) (d) Match the graph above with its Riemann Sum under. 5∑ i=1 f(xi−1) ∆x 5∑ i=1 f(xi) ∆x 5∑ i=1 f(x̄i) ∆x 5∑ i=1 f(x∗i ) ∆x 2. (6 pts) Circle the letter subsequent to every assertion that's at all times true. (a) ∫ cf(x)dx = c ∫ f(x)dx the place c is a continuing (b) ∫ [f(x)g(x)] dx = (∫ f(x)dx )(∫ g(x)dx ) (c) ∫ c a f(x) dx + ∫ b c f(x) dx = ∫ b a f(x) dx (d) ∫ f(x) g(x) dx = ∫ f(x)dx∫ g(x)dx (e) ∫ [f(x) ±g(x)] dx = ∫ f(x)dx± ∫ g(x)dx (f) ∫ b a f(x)dx = − ∫ a b f(x)dx / 10 pts 1 three. (1 pt) The operate f(x) = sin(x) is steady on what interval? (a) [−π,π] (b) [0,π] (c) [−π, 0] (d) (−∞,∞) four. (1 pt) Circle the phrase that greatest completes the assertion under: The equation ∫ x=1 x=−1 ln(x) dx = [x ln(x) −x]x=1x=−1 is (a) appropriate by the Basic Theorem of Calculus Half II. (b) incorrect as a result of the antiderivative of f(x) = ln(x) is F(x) = 1 x , not x ln(x) −x. (c) incorrect as a result of 1 x is discontinuous at x = zero. (d) incorrect as a result of ln(x) is discontinuous on [−1, 0]. 5. (2 pt every) Match the expression on the left with its definition on the precise. x̄i ∑n i=1 f(xi) ∆x ∆x xi−1 lim n→∞ ∑n i=1 f(xi) ∆x xi x∗i (a) a + i∆x (b) ∫ b a f(x) dx (c) ith subinterval midpoint (d) Internet space sum of n approximating rectangles (e) b−a n (f) ith subinterval pattern level (g) ith subinterval left endpoint 6. (2 pts) Think about factors C and D. One is the centroid of the shaded area; the opposite is the typical midpoint of n approximating rectangles as n → ∞. Label every level, and briefly clarify your decisions within the part under proper. / 18 pts 2 Integration 7. (10 pts every) Clear up each integrals: (a) ∫ [ 1 x3 + cos(2x) + 1 x ] dx (b) ∫ e √ x dx / 20 pts three eight. (10 pts) Select both (a) or (b) under to resolve. Don't resolve each. (a) ∫ 2π zero sin( 1 three x)2 dx / 10 pts four (b) ∫ 2x2+2 2x2+5x+three dx / 10 pts 5 Functions 9. (10 pts) Selected both (a) or (b) under to resolve. Don't resolve each. (a) A particle strikes alongside a line with velocity operate v(t) = t2−t, the place v is measured in meters per second. Discover the displacement and the gap traveled by the particle throughout the time interval [0, 5] / 10 pts 6 (b) Discover the centroid of the area certain by and f(x) = −x2 + four and the x-axis To obtain full credit score: 1) Graph the area under; 2) Find and graph the factors of intersection of f and the x-axis; three) Discover the coordinates of the centroid and graph it under. / 10 pts 7 Work A bucket that weighs four lbs and a rope of negligible weight are used to attract water from a effectively that's 80 ft deep. The bucket is full of 40 lbs of water and is pulled up at a price of two ft/sec, however water leaks out of gap within the bucket at a price of .2 lbs / sec. Discover the work finished in pulling the bucket to the highest of the effectively. 10. (1 pt) If F measures pressure and d distance, circle the equation under that measures Work (W ): (a) W = F d (b) W = Fd (c) W = F + d (d) W = F −d 11. (1 pt) The models of pressure appearing on the water and bucket because it rises to the highest of the effectively are (a) Newtons (N) (b) Joules (J) (c) Kilos (lbs) (d) Foot-Kilos (ft. - lbs) 12. (1 pt) The models of labor required to elevate the bucket and water to the highest of the effectively are (a) Newtons (N) (b) Joules (J) (c) Kilos (lbs) (d) Foot-Kilos (ft. - lbs) 13. (1 pt) Select the reply that greatest fills within the blanks. Because the bucket and water transfer up the effectively, the pressure appearing on bucket is as a result of its weight is , however the pressure appearing on the water is as a result of its weight is . (a) fixed, fixed, altering, altering (b) fixed, altering, altering, fixed (c) altering, fixed, altering, fixed (d) altering, altering, fixed, fixed. 14. (three pts) Based mostly in your reply to 13 above, calculate the work W wanted to haul the bucket alone from the underside to the highest of the effectively. / 7 pts eight Now, let’s determine the work required to haul the water all the best way up the effectively. 15. (1 pt) If t is measured in seconds, which expression is the same as the burden in lbs. of the water within the bucket at time t because it strikes up the effectively? (a) 40 .2t (b) 40(.2t) (c) 40 + .2t (d) 40 − .2t 16. (1 pt) If x is a measure of toes, which expression provides the gap of the water within the bucket from the underside of the effectively at time t? (a) x = 2 t (b) x = 2t (c) x = 2 + t (d) x = 2 − t 17. (2 pts) Use your solutions to questions 15 and 16 above to outline an expression for the burden of the water by way of x. Scratch work that will help you with Questions 15 and 16. Not obligatory to make use of, not counted for credit score! / four pts 9 To search out the work W required to haul the water 80 ft up the effectively, we begin by subdividing the 80 ft interval into n equally spaced subintervals of ∆x toes.-This part will discover the work required to haul the water up the effectively. We begin by subdividing the interval into equal components. When ∆x may be very small, the burden of the water is comparatively fixed all through the size of the interval. 18. (2 pts) Outline an expression that approximates Wi, the work required to elevate the water a distance of ∆x toes, from xi−1 to xi. By summing up all of the W ′is, we are able to approximate the work wanted to haul to water from the highest to the underside of the effectively. 19. (2 pts) Write an expression for the sum of all of the W ′is. We will make this approximation exact by letting n →∞ after which taking the restrict. 20. (four pts) Write an expression that calculates the exact quantity of labor wanted to haul the water from the underside to the highest of the effectively, after which resolve it. 21. (1 pts) Use your solutions to Questions 14 and 20 to calculate the full quantity of labor wanted to haul the water and bucket from the underside to the highest of the effectively.-Use the solutions to the questions under to calculate the quantity of labor wanted to drag the water and bucket from the effectively. / eight pts 10 Additional Credit score: (three pts) Which of the next areas are equal? To obtain credit score, your reply have to be justified by the proper clarification. (e) (f) / three pts 11 -research paper writing service
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