University:
Amherst College
Course:
MATH 106 | Calculus with Elementary Functions
Academic year:

2014
Views:

483

Pages:

17
Author:

ice0ver13toy

Math 106 Final Examination May 13, 2014 Answer Key 1. [30 Points) Compute each of the following derivatives. Ing (a) "(1), where f(x) = -luz(1) First, f'(x)= Next, f"(z) = Finally, f(1)= 1-Inz (-)-(1-x)(2) (2)2 -1-2(1-In 1) 1 --2(1-Inz) -1-2-3 d sec+ 244-4) عماع (9) Do not simplify the final answer here. First simplify using algebraic properties of the log. P R In sec+ elez (4-24)4 da --In (en (4-4) insecx+√ d = In (e) + In (3-2)+ sec+ -in (el)-in ((4-2)) In (secx+√x)-Inx-4ln (4-2) 1 --+ (+) (+)--()() 1 1 secx+√ 1 2 1 4- = (c) g(x), where g(x) = (sinx) Let y=(sinz). (or keep using y(x), it doesn't matter). Looking for Iny [(sin)]- In(sinx) = In 1 dz T Implicitly differentiate 1 dy 1 1 y da sin (cosx)+In(sinx)(2) Solve y dz 2900 zsing In(sinz) 2 (sinz) 500 rsin. 1 In(sin 2) = (d) dy if secy-cosre+(ln(c+5)) Implicitly differentiate: d dz = secy- dz 2 sec y(sec y tany)+ +(In(e+5))r +sinx= (+2))+n(+5) Distribute: 2 secy (secy tany) dy dy+sinx+2zye + In(e+5) dr Isolate dy dx terms: 2 sec cy(secytany) = 2xye+in(+5)-sinx da dr Factor: (2sec (2 secy tany- dz 2zye+ln(e+5)-sin z Solve: dy2zye+In(e+5)-sinz da 2 sec² y tan yery (e), where y= 1 1 1 + sin ve+e eve+Tuin 7 sinx+ First rewrite to simplify and prepare: Do not simplify. y= (sin ve+e) -1 +++7sinx+ (7 sloz+) = (sin ve+e) cos Vete 1 2+e (e) (continued...) ++7 1 2√2+7sin z (2+7 cos) (continued...) --(7sin z+17005+ 2 .2) حم(( 2. (20 Points) Compute each of the following derivatives. (a) f() where f(x) = sec(2x) + cos(2x)+ 2 sin First, f'(x)=2sec(2x) tan(2x)+2cos(2x)(-sin(2x))(2)-2(sinx) cos x. Next, f'()=2sec (2()) tan (2()) +2cos(2()) (sin (2())) (2) -- (()) 2 3 عام 16- (1+2+2)=1 16-1-2-d l3 Jo =15x-2-[(1513-23-3)-(0-2-)] =(1513-6-2)+2+ =1513-6-+2+]=[1513-4-1513-4-4][15]n3-8 15-2-da (d) Consider a different region bounded by y sinz, y=1, z=0 and bounded region. 2 Sketch the See me for a shaded sketch. It is the area bounded by the four curves below. Sketch: See Next Page. (e) Set-Up but DO NOT EVALUATE the integral to compute the volume of the three- dimensional solid obtained by rotating the region in (d) about the horizontal line y-1. Sketch the solid, along with one of the approximating washers. See me for a sketch. See Next Page. Volume= (outer radius)- (inner radius)) da = T 22-(1+sinx) dr (1) Consider a different region bounded by y Inz, y = 0, y 3 and 20. Sketch the bounded region. See me for a shaded sketch. It is the area bounded by the four curves below. (g) Set-Up hut DO NOT EVALUATE the integral to compute the volume of the three- dimensional solid obtained by rotating the region in (f) about the y-axis. Sketch the solid, along with one of the approximating disks. See me for a sketch. See Next Page. Note that y Inzz = e. Volume= (radius) dy =* (e) dy 10. [15 Points] Consider an object moving on a number line such that its velocity at time t seconds is given by u(t)=t2t feet per second. Also assume that the position of the object at one second is three feet. (a) Compute the acceleration function a(t) and the position function s(t). a(t)v(t)=2t-2 ft/sec o(t) = v(t) dt = -21 dt =-+C

Math 106 Final Examination 2014 Answer Key

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