Calculus I Practice Test Problems for Chapter 5 Page of 9 This is a set of practice test problems for Chapter 5. This is i o way a iclusive set of problems there ca be other types of problems o the actual test. The solutios are what I would accept o a test, but you may wat to add more detail, ad eplai your steps with words remember, I am iterested i the process you use to solve problems! There will be five problems o the test. Most will ivolve more tha oe part. You will have miutes to complete the test. You may ot use Mathematica or calculators o this test. Useful Iformatio that you will be give o this test: R b a i ( (b a) ) f a + i, L b a i ( (b a) ) f a + (i ). Determie a regio whose area is equal to the give limit. Do ot evaluate the limit. lim i. Cosider π iπ ta 4 4 ( ) d. (a) Compute the itegral by sketchig the itegrad ad computig the area uder the curve from the sketch. (b) Compute the itegral usig the Midpoit Rule ad two rectagles. Sketch the situatio ad eplai why the Midpoit Rule gives the eact aswer i this case. (. Evaluate the itegral ) d eactly by sketchig the itegrad ad iterpretig the itegral i terms of areas. 4. Evaluate (a) t cos( t ) dt (b) cos(l ) d 5. Evaluate (a) ( ) d (b) e t t dt 6. Evaluate (a) π/ cos + si d (b) ( ) d 7. Evaluate (a) e d (b) π/4 si θ cos θ dθ 8. A object moves alog a lie so that its velocity at time t (i m/s) is v(t) t 6t + 8. Fid the displacemet ad total distace traveled by the object for t 8. Displacemet to the right is positive. You may leave your aswer for the total distace traveled i the form: (Atiderivative) b a. 9. A cotiuous, odd fuctio f() has the property a (a) Sketch a graph to show geometrically why this is so. (b) Prove this statemet.. If f() is cotiuous o [, ], prove that a f() d f() d. f( ) d.
Calculus I Practice Test Problems for Chapter 5 Page of 9 Solutios Problem. Compare lim i π iπ ta 4 4 to the defiitio of the itegral usig right had edpoits i each subiterval: b a f() d lim i b a f(a + b a i) We eed to idetify a, b, f. Let s rewrite the origial epressio, ad see what we ca pull out of it: lim i π/4 ta π/4 i From this, we might guess that b a π 4 a b π 4 f() ta We ca check if our guess is right by substitutig: lim i π/4 ta(π/4 i) We have guessed correctly! A sketch of the area this represets is give below:
Calculus I Practice Test Problems for Chapter 5 Page of 9 Problem. I have sketched the fuctio y below. The itegral the curve, the shaded area. ( ) d is represeted by the area uder Sice the curve is above the -ais i the regio, we do ot eed to worry about subtractig areas to get the itegral. ( ) d shaded area (base)(height) ()() For part b, I have redraw my sketch, this time icludig the two rectagles that make up M (the poit iside each subiterval is chose to be the midpoit). ( ) d area of rectagles ()(.5) + ()(.5). This is the actual area sice the area beig missed is equal to the area beig overestimated for each rectagle.
Calculus I Practice Test Problems for Chapter 5 Page 4 of 9 Problem. The itegrad is y, which simplifies to (y + ) +, which is a circle of radius cetered at (, ). We oly wat the bottom half of the circle sice we bega with. The itegral is the shaded area, which is the area of the rectagle with width ad height plus oe quarter the area of a circle of radius. We also have to make this egative sice it is below the -ais. ( ) ( d ()() + 4 π()) π 4. Problem 4a. t cos( t ) dt Substitute: u t du t dt t cos( t ) dt cos( t )t dt cos( t ) ( t dt) ( ) cos u du si u + c si( t ) + c Problem 4b. cos(l ) u l d Substitute: du d cos(l ) d cos(l ) d cos u du si u + c si(l ) + c
Calculus I Practice Test Problems for Chapter 5 Page 5 of 9 Problem 5a. ( ) u d Substitute: du ( ) d ( ) d ( ) d udu u / du u / (/) + c ( ) / + c Problem 5b. Problem 6a. π/ π/ Problem 6b. t et dt Substitute: t et dt e t dt t e u du e u + c cos + si d e t + c cos + si d u t du t / dt t dt Substitute: π/ u + si du cos d Chage limits: whe, u whe π, u + si cos d u du u / du u/ (/) ( ) ( d Substitute: ) u du d Chage limits: whe, u whe, u 8
Calculus I Practice Test Problems for Chapter 5 Page 6 of 9 ( ) d ( d ) 8 8 u du u du ( ) u 8 ( ) [ ] 8 u [ ] 8 [ 4 8 ] 4 [ 5 ] 5 4 48 Problem 7a. Problem 7b. π/4 π/4 e d e d si θ cos θ dθ Substitute: u du d Chage limits: whe, u whe, u e ( ) d e u du [eu ] [ e e ] [ ] e Substitute: si θ π/4 cos θ dθ / / u cos θ du si θ dθ Chage limits: whe θ, u whe θ π/4, u cos π/4 ( si θ)dθ cos θ du u u du
Calculus I Practice Test Problems for Chapter 5 Page 7 of 9 ( u ( ) ( ) / u / ) / Note: If you forget that cos π/4 /, draw a little uit circle ad work out the value usig geometry. Problem 8. displacemet s(8) s() 8 8 ( t v(t) dt (t 6t + 8) dt ) 8 t + 8t ( ) 8 (8) + 8(8) ( + ) 5 9 + 64 8 m The particle moved 8/ m to the right (because the aswer was positive). distace traveled 8 v(t) dt We eed to work out the absolute value. I this case (quadratic), it is easiest to do by fidig the roots of the quadratic. If you ca t fid the factorizatio by ispectio, you ca use the quadratic formula. v(t) t 6t + 8 (t 4)(t ) Sice v(t) is a parabola opeig up, we kow that it must look somethig like the followig: From this graph, we ca figure out whe the fuctio is positive, ad whe it is egative.
Calculus I Practice Test Problems for Chapter 5 Page 8 of 9 This allows us to rewrite the itegral as follows: distace traveled 8 8 ( t v(t) dt t 6t + 8 dt t 6t + 8 dt + (t 6t + 8) dt t + 8t ) 4 4 t 6t + 8 dt + (t 6t + 8) dt + ( t t + 8t ) 4 8 4 8 4 t 6t + 8 dt (t 6t + 8) dt ( t + t + 8t ) 8 4 Problem 9. Sice f() is odd, we kow that f( ) f(). This meas the fuctio must be symmetric with respect to rotatio of 8 degrees about the origi. Graphically, this meas the fuctio looks somethig like: The shaded area A is equal to the shaded area A. However, A is below the ais. Therefore, the two areas cacel i the itegratio, ad a a f() d. To prove this result, we eed to work with the itegral ad do a substitutio: If f is odd, the f( ) f(). a a f() d a a f() d + a a a f() d + f( u) du + a f(u) du + f() d + a f() d a a a f() d f() d First Itegral Substitutio: f() d (sice f is odd) f() d (Substitutio: u (othig will chage)) u du d Chage Limits: whe u whe a u a
Calculus I Practice Test Problems for Chapter 5 Page 9 of 9 Problem. f() d f() d Substitute: f() ( d) f( u) du u u du d d du Chage limits: whe, u whe, u f( u) du (Substitutio: u (othig will chage)) f( ) d