[answered] Math 153 FINAL TH PORTION Name (Last, First): 75 pts SCORE

Calculus III (Community College Level) Final Review sheet. I wasn't able to grab the answer sheet in class, so I have no idea if the answers and methods I'm using are correct.

This sheet covers Power Series, Function Rep. of Power Series, Taylor/MacLaurin Series, Vector Functions (in regards to position, speed, and acceleration), and parametric equations of Planes/Lines.

Please complete the worksheet and show ALL WORK. Like, every single step as if you were teaching it to someone. Small algebra calculations can be done in head and not required to write down.

Math 153

 

FINAL TH PORTION Name (Last, First):

 

75 pts SCORE (1.) (a.) Derive a power series for the function f (x) = cos(x3 ) (b.) Determine the interval of convergence. Z 0.1 (c.) Use 5 terms to give a 6 decimal place approximation of cos(x3 )dx. Show all of your work. ?0.1 (d.) How does this compare to the answer obtained using your calculator? 1 2 

 

2

 

3t

 

2

 

?

 

, te , sin (4t) describe the velocity of a particle at time t. (show your work

 

16 ? t2

 

on all derivatives and integrals.) (2.) Let #?

 

v (t) =  (a.) Determine the acceleration at time t. (b.) Determine the position at time t. 3 (3.) Let #?

 

r (t) =  t2 t3

 

, ,

 

2 3 Z 3 

 

dt 0 #?

 

(a.) Determine T(t), simplify fully and show all work. #?

 

(b.) Determine N(t), simplify fully and show all work. #?

 

(c.) Determine B(t), simplify fully and show all work. 4 (4.) Let f (x, y) = 9 ? x2 ? y 2

 

(a.) Determine the

 

parametric equations of the line tangent to the surface in the x direction at the point

 

1, 1, f (1, 1) (b.) Determine the

 

parametric equations of the line tangent to the surface in the y direction at the point

 

1, 1, f (1, 1) (c.) Determine the parametric equations of the line that is normal to the surface at the point 1, 1, f (1, 1) (d.) Determine all points at which the line from part (e.) intersects the surface.

 

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