1. Find the derivative of the following functions. a) f(x) = 2 + 3x b) f(x) = (5 2x) 8 c) f(x) = e2x

Transcription

1 I. Dierentition. ) Rules. *product rule, quotient rule, chin rule MATH 34B FINAL REVIEW. Find the derivtive of the following functions. ) f(x) = 2 + 3x x 3 b) f(x) = (5 2x) 8 c) f(x) = e2x 4x 7 +x+2 d) f(x) = (3 ln x)(2x 2 + ) e) f(x) = sin(3x 2 ) 2. Find the slope of tngent to f(x) = cos(2π x) + tn(2x) t x = π. b) Applictions. *derivtive is instnteneous rte of chnge *rst derivtive positive (negtve) mens originl function is incresing (decresing) *nd where derivtive is zero to nd mximum or minimum of function 3. After t yers, the number of rbbits in forest is given by the formul N(t) = 2t 2 5t +. ) How mny rbbits re there originlly? b) At wht rte is the number of rbbits chnging fter 2 yers? c) Is the number of rbbits incresing or decresing fter 2 yers? d) When will the number of rbbits be minimum? 4. The totl monthly revenue of compny is given by the formul R(q) = 525q.3q 2 dollrs when q units re produced nd sold per month. The compny currently produces nd sells units ) Use mrginl nlysis to estimte the chnge in revenue if the compny produces nd sells units insted. b) Bsed on your nswer in prt ), would you recommend the compny to increse the level of production to units? c) Use R(q) to compute the ctul chnge in revenue when the number of units produced nd sold is incresed to. Does it gree with your recommendtion in prt b)? d) If the compny wnts to generte the mximum mount of revenue, how mny units should it produce nd sell? II. Integrls. ) Indenite integrls. *dierentition bckwrds 5. Find the following indenite integrls simply by thinking bckwrds. ) x 4 b) x c) 2xe x2 d) sec 2 x e) sin x

2 b) Denite integrls. *signed re under the grph nd bove the x-xis MATH 34B FINAL REVIEW 2 6. Find the following denite integrls by grphing the function nd interpreting the integrl s the signed re between the grph nd the x-xis. ) 5 e b) 3x c) 5 x 2 d) x2 (Hint: Wht kind of curve is this?) c) Riemnn sum *use rectngle to pproximte the re under the curve 7. ) Approximte 4 2x2 using right-hnd Riemnn sum with 4 subintervls. b) Approximte 2 e x using left-hnd Riemnn sum with 6 subintervls. d) Fundmentl Theorem of Clculus i. Prt I If F is dened to be F (x) = ˆ x c f(t)dt, then F is dierentible nd F (x) = f(x). Furthermore, if g(x) is function of x nd then by the chin rule G(x) = F (g(x)) = ˆ g(x) c f(t)dt, G (x) = F (g(x))g (x) = f(g(x))g (x). 8. Find the derivtive of the following functions. ) F (x) = x 4 sin(t2 + π)dt b) G(x) = x 2 ln(2t + )dt c) G(x) = e x t2 + dt (Hint: How do you ip the bounds so e x is the upper bound?) ii. Prt II If F (x) = f(x), then 9. Find the following denite integrls. ) π 2 cos x b) 9 4 x(2x x 2 + ) c) (ex x) d) Methods of integrtion. i. U-substitution. f(x) = F (b) F ().. Find the following indenite integrls using u-substitution. ) sin(4x) b) xe 5x2 c) ln x 2x. Find the following denite integrls using u-substitution. c) 2 4x+2 x 2 +x+ d) x 3 +6x x 2

3 MATH 34B FINAL REVIEW 3 e) (3 + 5x x2 ) (2x 5) ii. Integrtion by prts. *The formul for integrtion by prts is ˆ ˆ udv = uv vdu 2. Find the following indenite integrls integrtion by prts. ) 2x ln x b) xe 3x c) cos( x) (Hint: Use u-substitution on x rst nd the use integrtion by prts.) 3. Find the following denite integrls integrtion by prts. ) π x cos x b) 4 x2 ln x (Hint: Use integrtion by prts twice.) iii. Prtil frctions. 4. Find the following indenite integrls using prtils frctions. ) x 2 b) x 2 2x 5. Find the following denite integrls using prtils frctions. ) 3 2 x 2 +5x 6 b) x x 2 3x+2 e) Applictions. i. Ares. 6. Drw the region bounded by the given curves nd nd its re. ) y = x 2, y =, x = b) y = x, y = x c) y = x, y = x, x = 3 (Hint: Use two integrls to nd the re of ech region.) d) y = e x, x =, y = e) y = sin x for x π, y = π 4 ii. Volumes. The volume of the solid obtined by rotting the region under the grph of y = f(x) for x b bout the x-xis is π f(x) 2 You should think of f(x) s the rdius of typicl disc nd is its thickness. 9. Let R be the region under the grph of y = 8 for x 4. ) Use integrl to nd the volume generted by rotting R bout the x-xis. b) The volume generted is in fct cylinder. Wht is its rdius? Wht is its height? c) The volume of cylinder is given by the formul V = πr 2 h. Use this to check your nswer in prt ). 8. Let R be the region under the grph of y = 4x for x. ) Find the volume generted by rotting R bout the x-xis. b) The volume generted is in fct cylindricl pyrmind. Wht is its rdius? Wht is its height? c) The volume of cylindricl pyrmid is given by the formul V = 3 πr2 h. Use this to check your nswer in prt ). 9. Let R be the region bounded by the grph of y = x nd the x-xis. Find the volume generted by rotting R bout the x-xis.

4 iii. Dierentil equtions. MATH 34B FINAL REVIEW 4 *Integrte to nd the originl function nd then use the given point to nd the constnt C. 2. Solve the following dierentil equtions. ) dy = e x 6 +, y() = 9 b) df = 2 5 3x, F () = iv. Some word problems. *The Fundmentl Theorem of Clculus tells us tht if we integrte f (x), we get f(x) + C bck. To nd C, use point tht is given. *The Fundmentl Theorem of Clculus lso tells us tht f (x) = f(b) f(). So, integrting the rte of chnge over [, b] gives us the overll chnge of the function f from x = to x = b. 2. A bll is thrown upwrd from the top of building with initil velocity 55ft/sec. Recll tht ccelertion due to grvity is 32ft/sec. ) Wht is the velocity of the bll t seconds fter it is thrown? b) Wht is the chnge in position of the bll from t = to t = 3? c) If the building is 4ft tll, how high bove the ground is the bll t t = 3? d) When will the bll hit the ground? 22. The price of stock is $3 per shre in Jnury 2. If its rte of chnge t months fter Jnury 2 is given by the formul r(t) =.8t +. ) Wht is the price of the stock per shre t months fter Jnury 2? b) When will the stock become worthless? c) In order to mke the mximum mount of prot, when should you sell the stock? v. Averge vlue. *The verge vlue of function on the intervl [, b] is given by b f(x) 23. The price p in dollrs for product is chnging t the rte of p (x) = x 2x2 + 3 when x hundred units re demnded by consumers. The price is $9 when 6 units re demnded. ) Find the demnd price function p(x). b) At wht price will units be demnded? c) How mny units will be demnded when the price is $? 24. The velocity of cr is given by the function v(t) = 2t 2 + t. ) Wht is the chnge in the cr's position from t = to t = 3? b) Wht is the verge velocity of the cr over the time intervl [, 3]? c) When is instntneous velocity equl to the verge velocity you found in prt b)? Find such t tht lies in [, 3]. 25. The rte of chnge of temperture (in degree Celcius) of freshly bked cke is given by the function T (t) = t + 4 for t 3, where t is the number of hours fter the cke hs been tken out from the oven. ) Find the overll chnge in temperture over the time intervl [, 3]. b) Wht is the verge rte of chnge of temperture in [, 3]?

5 MATH 34B FINAL REVIEW 5 c) If the originl temperture of the cke ws 6 degree Celcius, wht is its temperture 3 hours fter? d) If you wnt to et the cke t temperture of 4 degree Celcius, when should you et it? 26. A compny sells the q-th unit of product t price given by the function p(q) =.3q 2 + q + 4 in dollrs. So, for exmple, the price of the st unit is p() = 4 while the price of the th unit is p() = 2. ) Wht is the totl cost for the rst units? (Hint: The cost for the q-th unit is p(q). Sum ll of the p(q) up using integrl to get the totl cost.) b) Wht is the verge cost for the rst units? III. Miscellneous. 26. π cos2 x 27. Suppose we re given tht f(x) = 4, Find the following denite integrls. ˆ 2 2 f(x) = 2, 3f(x) 2g(x) nd ˆ 2 g(x) =, nd 5f(x) + 2g(x). 2 g(x) = Consider the function f(x) = x 2. ) Sketch the function on the intervl [, 5]. b) Wht is the mximum vlue of f(x) on this intervl? Wht is the minimum? c) Using property of integrls nd prt b), give lower bound nd n upper bound for the integrl x 2.