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# Calculus III Study Guide MATH 2243

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This 7 page Study Guide was uploaded by Joy Bullington on Sunday October 2, 2016. The Study Guide belongs to MATH 2243 at Georgia Southern University taught by Dr. Stone in Fall 2016. Since its upload, it has received 6 views. For similar materials see Calculus III in Math at Georgia Southern University.

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Date Created: 10/02/16

Memorize: Curves in Space: ^(t=¿cost ,sint,t>isaHelix r(t=postion ' ´ t)=velocity ' ' ´ (t=acceleration speed=¿r ´'t)∨¿ Velocityvector=Tangentvector accelerationvector points→centerof ¿ellipse ˇ product of v∙a ¿seeif itisorthoganol(v∙a=0) Projectile Motion: v0 t= sinѲ g v sin ϴ ymax= 2g x a 1 1 −1 2 2dx= sec (¿)+C x √ −a a ∫ ¿ b ArcLength= sp∫ed dt a ¿v(t)∨¿ ´(t) UnitTangentVector= ¿ Curvature: ¿r ∨¿= 1 raduis ´ ' ¿ ¿T ∨ ¿ κ=¿ ' ¿T ∨¿ T ' N= ¿ B=T X N ´ Bis⊥T∧⊥N ´ R λ scaling factor)= r λ = (ratioof radius) a = d ∨V∨¿ T dt 2 aN=κ∨V ∨¿ ' ' ¿ ¿ f ( )∨ 23/2 {1+[f (x)]} κ ( )¿ Partial Derivatives: ¿,>¿ Aregionisopenif ithasnoboundary points¿ Regionisclosedif itcontainsallof itsboundary points(≤,≥) Interior Pointif youcandrawaaroundit∧itisstill∈region Boundary pointif itis both∈¿out of region Aregionisboundedif itis finite ∂ f (x,y) =∂withrespect¿x ∂x ∂ f (x,y) =∂withrespect¿y ∂y ∂z = f xx,y )=z x ∂x Second Partials: ∂ f Second∂ Derivative= 2 ∂x 2 MixedSecond∂= ∂ f ∂y∂x 2 2 ∂ f ∂ f ∂y∂x = ∂x∂y fxy(a,b = f yx,b) ∂Derivativestellustherateof change∈acertaindirection slope= f a+Δx )−f (a) Δx tangentlane=limitof allsecantlines slopeof tangent=lim f(a+Δx − f (a) Δx →0 Δx Questions: Curves: Find velocity and acceleration vectors at time t. t 1 −1 1. r(t= t+1 i+ t j ,t= 2 t 2 2t 2. r(t=e i+ e9j ,t=ln3 r(t= (cos2t i+ 3sin2t )j ,t=0 3. Find velocity and acceleration vectors at time t, and find speed. r(t= (+1 )+ t −1 )j+2tk ,t=1 4. t2 t3 5. r(t= (1+t)i+ j+ k,t=1 √2 3 t = 2cost )i+(3sint)j+4t k ,t= π 6. r 2 t = (secti+ tant)j+ t k,t= π 7. r 3 6 2 t)= 2ln (t+1 )i+t j+ t k ,t=1 8. ( ) 2 r(t= (e−ti+(2cos3t) j+(2sin3t)k ,t=0 9. Find the angle between the acceleration and velocity vectors. 2 10. r(t= (3t+1 i+ √t j+t k ,t=0 Evaluate Integral 1 3 11. 0 [ i+7 j+ (t+1)k]dt π 4 12. ∫ [sint)i+(1+cost) jsec tk ]t −π 4 4 ∫ 1 i+ 1 j+ 1 k dt 13. 1[t 5−t 2t ] Solve the initial value problems for r as a vector function of t. dr =−ti−t j−tk ,r 0 =i+2 j+3k 14. dt dr =(180t)i+(180t−16t ) j ,0 =100 j 15. dt dr = 3(t+1)1/i+e j+ 1 k ,( ) =k 16. dt 2 t+1 2 d r =− (i+ j+k,r(0=10i+10 j+10 k∧dr ¿t=00 17. dt2 dt 18. A projectile is fired at a speed of 840 m/sec at an angle of 60 degrees. How long will it take to get 21 km down range? 19. A projectile is fired with an initial speed of 500 m/s at an angle of elevation of 45 degrees. a. When and how far away will the projectile strike? b. How high overhead will the projectile be when it is 5 km downrange? c. What is the greatest height reached by the projectile? 20. A baseball is thrown from the stands of 32 feet above the field at an angle of 30 degrees up from the horizontal. When and how far away will the ball strike the ground if its initial speed is 32 feet/sec? 21. A spring gun at ground level fires a golf ball at an angle of 45 degrees. The ball lands 10 meters away. a. What was the ball’s initial speed? b. For the same initial speed, find the two firing angles that make the range 6 m. 22. An electron in a TV tube is beamed horizontally at a speed of 5X10 m/s toward the face of the tube 40 cm away. About how far will the electron drop before it hits? 23. A gold ball is hit with an initial speed of 116 feet/sec at an angle of elevation of 45 degrees from the tee to a green that is elevated 45 feet above the tee as shown in the diagram. Assuming that the pin 369 feet downrange, does not get in the way, where will the ball land in relation to the pin? Arc Length Find the unit tangent vector and the length of the portion of the curve. 1. r( ) (2cost) ( 2sin) j√ 5t k,0≤t≤π 2. r(t= (6sin 2)i+6cos2t )j+5t k ,0≤t≤π 3/2 r(t=ti+ 2 k,0≤t≤8 3. (3 t sin ¿ 4. ¿ r( )=(cos ) j+¿ Curvature Find unit tangent vector, N and curvature. r(t=ti+ (lncost)j ,− <t< π 1. 2 2 r(t=(2t+3)i+ 5−t 2)j ,− <t< π 2. 2 2 3. r(t= (cost+tsini+ sint−tcost)j+3k Partial Derivatives Find values 1. f(x ,y=sin (xy) f (2, ) a. 6 f (−3,π ) b. 12 1 c. f (π,4) −π d. f (2 ,−7) Find and sketch domain for each function. 2. f(x ,y)=√y−x−2 3. f(x ,y=ln (x +y −4 ) (x−1 )(y+2 ) 4. f(x ,y= 3 (y−x)(y−x ) sin(xy) 5. f(x ,y= 2 2 x +y −25

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