Math 152 Take Home Test 1

Transcription

1 Math 5 Take Home Test Due Monday 5 th October (5 points) The following test will be done at home in order to ensure that it is a fair and representative reflection of your own ability in mathematics I expect you to comply with the following conditions.. I will not talk to any person about any part of this test either directly or indirectly.. I will not use the internet or any mathematic program such as Scientific Notebook.. I can use my class notes; solutions manual; text book; homework assignments and graphing calculator. In order to make this test a fair reflection of my ability I promise to comply with the above conditions. Students Signature: Print Name:

2 . Estimate the value of the integral Do all calculations to decimal places. x + dx by finding L, R and M. Use the Fundamental Theorem of Calculus (Part ) to find g (x). (a) x g( x) t tan tdt (b) g(x) sin x tdt

3 . Find the following indefinite integrals, you must do so algebraically and show the relevant working. (a) (x + e x x + x e )dx (b) e x dx x dx (c) xx 5 (d) (sec sin ) d

4 . Evaluate the following integrals, you must do so algebraically and show the relevant working. (a) x 8x dx x (b) dx x

5 5. Use the substitution method to calculate the following indefinite integrals. dx (a) xx (b) e sin cos d (c) x dx x 5

6 6. Use the substitution method to calculate the following indefinite integrals. (a) x x dx Use u x (b) ex dx e x (c) sec x tan x dx

7 7. For each of the following integrals do not evaluate them only state the substitution that you would use to find the integral. (a) (x + )(x + x) dx Substitution u x + x (b) cos θ sin θ dθ Substitution u sin θ (c) x x + dx Substitution u x + (d) tan x sec x dx Substitution u tan x or u sec x (e) sin x x dx Substitution u sin x 8. The acceleration function a(t) t + and v() 5, S() where t (a) Find an expression in terms of t for the velocity function v(t). (b) Find an expression in terms of t for the displacement function s(t). (c) Find the distance traveled from t to t.

8 9. Find the Area between the two curves in the Diagram Shown

9 . Find the Area between the two curves in the Diagram shown

10 . Find the area between the following two curves, y sin x, y cos x,x, x. Find the Volume of the solid created by rotated the curve y x about the y-axis from y to y.

11 . Find the volume using the Disk Method, the region given below.

12 . Find the exact volume of the solid obtained by rotating the region bounded by the curves y x, y, x about the line y

13 . Estimate the value of the integral Do all calculations to decimal places. Solutions x + x dx by finding L, R and M b a n x f(x).5.. x + L x(f() + f() + f() + f()) L ( ) L.8 R x(f() + f() + f() + f()) R ( ) R x f(x) x x(f(.5) + f(.5) + f(.5) + f(.5)) M ( ) M. M Use the Fundamental Theorem of Calculus (Part ) to find g (x). (a) x g( x) t tan tdt (b) g(x) tdt sin x g (x) x tanx (t t sin x ) t ( t t sin x ) t g(x) sin x () sin x g (x) sin x cos x g (x) sin x cos x

14 . Find the following indefinite integrals, you must do so algebraically and show the relevant working. (a) (x + e x x + x e )dx (x + e x x + x e )dx x + e x + x + x xe + C x + e x + X + x xe + C (b) e x dx x e x + ln(x) + C dx (x 5x)dx x 5 x + C (c) x x 5 (d) (sec sin ) d sec θ + ( cos θ) dθ sec θ + cos θ dθ tanθ + θ sinθ + C. Evaluate the following derivatives, you must do so algebraically and show the relevant working. (a) 8x x dx x x x x x ( + () ) ( + ) 6 x (b) x dx dx x x x x dx x x x x x x x x () 5 6 () 7 7

15 5. Use the substitution method to calculate the following indefinite integrals. dx (a) xx xu u du du x Substitution u x du dx x u c () du x.dx (x ) + C du x dx (b) e sin cos d e u cos du Substitution u cos e u du du d e u + c du e sinθ + C du cos sin cos cos d d 5.(c) x dx x 5 x x 5 dx Substitution u x 5 x u u x du du du dx du x x dx u c x 5 c x du dx x 5 + C

16 6. Use the substitution method to calculate the following indefinite integrals. (a) x x dx x u du Substitution u x so (x u +) u+ u du du dx (u + )u du du dx (u + u )du u + u + C u + u + C x x dx (x ) + (x ) + C (b) ex e x dx ex u e x du Substitution u ex ex u du du ex dx e x u du u du u + C e xdu du ex dx dx u + C ex dx e x ex + C (c) sec x tan x dx sec x u sec x du Substitution u tan x u du du dx sec x u + C du sec x dx sec x tan x dx tan x + C sec x du dx

17 7. For each of the following integrals do not evaluate them only state the substitution that you would use to find the integral. (a) (x + )(x + x) dx Substitution u x + x (b) cos θ sin θ dθ Substitution u sin θ (c) x x + dx Substitution u x + (d) tan x sec x dx Substitution u tan x or u sec x (e) sin x x dx Substitution u sin x 8. The acceleration function a(t) t + and v() 5 where t (a) Find an expression in terms of t for the velocity function v(t). v(t) a t) dt ( t dt t + t + C Since v() 5 we can use this to find the value of c v() () + () + C 5 C 5 So v(t) t + t + 5 (b) Find an expression in terms of t for the displacement function s(t). s(t) v ( t) dt t t 5 dt t + t + 5t + C (c) Find the distance traveled from t to t. Distance traveled from t to t is equal to t t 5 t t t 5dt t t 5t t () () 5() dt

18 9. Find the Area between the two curves in the Diagram Shown Area cos x sec xdx ( cosx ) sec xdx ( cosx ) sec xdx cosx sec xdx x { x sinx tan x } x { π + sin π tan π ) ( + sin tan)} ( π + sin π tan π ) Area ( π + )

19 . Find the Area between the two curves in the Diagram shown Area cos y ( sin y 6 cos y sin y 6 sin y cos y dy y y 6 dy (sin π cosπ) (sin ( π 6 ) cos ( π )) ( + ) ( ) Area 9

20 . Find the area between the following two curves, y sin x, y cos x,x, x Area cos x sin x dx sin x cos x x x (sin cos ) (sin cos) ( ) ( ) Area. Find the Volume of the solid created by rotated the curve y x about the y-axis from y to y. b Volume π(f(y)) dy a π (y ) π y dy dy π ( 5 y5 y y ) π ( 5 y5 y y ) π ( 5 5 ) () π ( ()) () 5 Volume 5 π

21 . Find the volume using the Disk Method, the region given below. b Volume (( f ( x)) ( g( x)) ) dx a (( x ) ( x) ) dx ( x x ) dx x { x x } π ( ) () x π

22 . Find the volume of the solid obtained by rotating the region bounded by the curves y x, y, x about the line y (Using the disk method) y Small Big y x y y x Typical cross section cut b Volume (Big) (Small) dx a ( x ) () dx 6 ( x x ) 6 x x 6 x x 6 x x dx x x x 7 x 7 x 7 7 () () () () () () Volume 5 7 π