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midterm review - Math 242 Midterm — Review(W10 1 Evaluate...

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Unformatted text preview: Math 242 Midterm —- Review (W10) 1. Evaluate each integral. af(:-;1—+ +2x)dx b.f 5: dx of” 11.14“” 4x2+7 ex2+1 e. f3(lnx)3 dx f. f Sxdx g. I4xdx h. f31:x(x)dx 2x (2x+3)2 92""1 2. The marginal cost of producing the xth case of toothbrushes 15 x3 (100 + x4) and the fixed cost is $1000. Find the cost fimction C (x). 3. A marginal revenue function is determined by R’(x) = axe bx , where a and b are non-zero real constants. Find the revenue function R(x) in terms of a and b if R(0) = 0. 4. Estimate the value of f: m for n= 4 using a) a lefi: Riemann sum, b) a right Riemann sum, 1 x2+3 and c) midpoints. 5. Suppose the rate of sales of an item is given by f (t) = —3t2 + 36 items per week, where t is the number of weeks after an advertising campaign has begun. How many items where sold during the third week? 6. It is estimated that t years from the beginning of the year 2000, the demand for oil in a certain country will be changing at the rate of (1 + 2t)"1 billion barrels per year. Is more oil consumed (demanded) during 2001 or during 2004? How much more? 7. Find the area of the region bounded by a. f(x) =—x and g(x) = 2—x2 from x= ~—1to x= 3. b. f(x) = 2x2 and g(x) = x3 — 3x. Find the consumers’ surplus for the demand equation p = 29 — q2 at a market price of p = 4. 9. The demand and supply equations for a certain commodity are given by 50p + q = 600 and 20p = q.+ 100. Find the equilibrium point, and then calculate the consumers’ surplus and the producers’ surplus. 10. The total revenue in thousands of dollars of a certain firm is given by R06) = xV9 + x2 where each x represents 1000 items sold. Find the average total revenue for the first 4000 units sold. 11. If the average value of the function f (x) —-——- over the interval 0 < x < 2 is 3‘ Determine the value for a, if a is a positive real constant.+ 12. If f (x) is continuous on the interval [—2, 8] where I: f (x) dx = 25 , and I: f (x) dx = —8 , compute the value for fBSH + 2 f (x)] dx . 13. The following graph shows the derivative f ’(x) of some function f (x) and the values of some areas. a. Find In“ f'(x) dx . b. What is the average value of f ’(x) over the interval [0, 5]? c. If f (0) = 10, sketch a graph of f (x) and give the coordinates of the local maxima and minima. 9° Area 7—.‘20 - 14. Suppose money is transferred continuously into a market account at the constant rate of $16,000 per year. Determine the balance in the account at the end of 4 years if the account pays 8% annual interest compounded continuously. 15. Find the 6111011181? 5011111011 01 1116 11/611 uifferenual 6"uat1"1‘1 ‘1'11’1 S" 1131 165 the 13/611 COfidlfiOD. 1’ H x dy xy __ fl-Lw _. _ dx=x2+1 where y— 3whenx—0 b.dx—3y2w erey—Z when x—O a. l l l l l 1 1 , v i KEY la 51n|x1—2—‘;+§x8+c b. Let u=2x2+3 —) —§(2x2+3)-1+c c. Letu=4x2+7 —» 31n(4x2+7)+c (1. Let u = x2 + 1 —> —2e”‘2’1 + C e. Let u = ln(x) —> §[ln(x)]4 + C f. u=5x; dv=(2x+3)_2dr 1 —1 5 —1 5 1 5 —r 5 du=5d\:; v=—7(2x+3) 2—7x(2x+3) +j —dx=—§x(2x+3) +Iln|2x+3I+C 2x+3 NOTE: This integral may also have been calculated by substitution. Eg. Let u = 2x + 3 etc. Try it, the answer will appear different. Check by taking the derivative! -2x+1 g. u = 4x ; dv = e 1 — _ ; du = 4d): ; v = ~§e 2‘“ => —-2xe 2"“ + 2]. e'2‘+'dx=—2xe‘2“’l—e”2"‘“+C g 3 -3 h u=ln(x),dv=7xdx f d_1dx' 3.2 3.21 314(1): 3.21n 3.2C 3 11—; , v——Zx 3—K): n(x)+z x ——2—x (x)—-§x + 2. C(x) = 25x4 +§x8 + 1000 a a a 3. R(x) = zxebx ~b—zebx +35 4 a. as 1.199 b. z 1.199 c. ”1.214 3 5. f(—312+36)dt = [—13 + 36!]: =17 items 2 2 5 . 6. More oil was consumed during 2001 I (1+21)‘1 d! :1 0.2554 billion barrels gs. {(1-121)‘l d: x 0.1003 billion barrels in 2004; a difference 1 4 of about 0.1551 billion barrels more. 2 3 . l9 7 a —x = 2 — x2 :5 x = 2 or x = -1 Graph both functions! :> Area= [(—x2+x+2)dx + J.(x2~—x—-2)dx = ? —1 2 0 3 b. 2;:2 = x3 — 3:: :> x = -1, 0,3 Graph both functions! => Area= J-(x3—2x2—3x)dx + J'(—x3+2x2+3x)dx=11,8§ -I 0 5 8. CS = I(29—qZ)dq — 20 z $83.33 0 9. 12 — 0.02q = 0.054 + 5 => Equilibrium at (q, p) = (100, 10) 100 100 CS = I (12—0.02q)dq-1000=$100 ; PS= 1000— I (0.05q+5)dq = $250 0 0 §. 1 3 .. i 10 ll): 9+x2dx z $8 16667 i . 4 0 , . l 11 a - 2—1 i . — 2 g 12. -78 j 133 6 b -1 2 i C. 0.094 4 14. FV = e L 16,oooe*’-°8'di 2 $75, 426 15 a. y=—3Vx2+1 b. y=3ex+7 ...
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