(1) Sketch the curve represented by the vector-valued function, and give the orientation of the curve as well:

a)r(t)=(t1)i+t2j

b)r(t)=2costi+3sintj

2) For r(t)=ti+t3jSketch the curve, and the velocity and acceleration vectors at t = 1 (on the same set of axes).

3) Find the limit:limt[(t2+1t)i+(4t23t2)j+(e2tt)k]

4) Solve the differential equation:

r(t)=e2ti+1+t22tj+3t2k,r(0)=2i3j+4k

5) A baseball, hit 3.5 feet above the ground, leaves the bat at an angle of 45 and is caught by an outfielder 3.5 feet above the ground, 250 feet away from home plate. What was the initial speed of the ball, and what was it's maximum height?

6) Determine the maximum height and range of a projectile fired at a height of 5 feet above the ground with an initial velocity of 1100 feet per second and at angle of 30 above the horizontal.

7) Find the principal unit normal and tangent vectors for the vector-valued function at

t=1: r(t)=(2t)i+t2j

8) Find the tangential and normal components of acceleration at t= 2 for the function

r(t)=ti+t3j

9) Find the tangential and normal components of acceleration for

r(t)=2cos(t)i+2sin(t)j

10) Find the curvature of the function

r(t)=ti+21t2j+31t3k

11) Find the curvature in rectangular coordinates:

y=3x24

12) Find the slope of the functionf(x,y)=xe3xyin the x and y direction at the point(1,0,e)

13) Find the four second partial derivatives of f, and verify that the mixed partials are equal.

f(x,y)=x34x2y2+y3

14) Find the gradient of the functionf(x,y)=e2xcos(y), and the directional derivative in the directionofu=21i21j at (0,0,1)

15) For the functionf(x,y)=2xyx+y

(a) describe the domain of the function

(b) discuss it's continuity

(c) Assesslimx(2,2)f(x,y)

16) What is the direction of maximum increase on the surfacef(x,y)=x34x2y2+y3from the point (1,1,-2)

17) Find the directional derivative off(x,y)=x2siny @ (2, 0) in the direction ofv=2i+2j (you might want to check v to see if it's a unit vector).

18) Find the gradient off(x,y)=x2+xy+y2 and use the gradient to give the directional derivative in the directionPQ ,where P= (1,1), Q= (5,4)

19) For the functionf(x,y)=yeyxfind the direction of maximum increase from the point (2,1).

20) Find the equation of (a) the line normal to, and (b) the plane tangent tof(x,y)=x2y2at the point (2,1,4)

21) Use the second partials test to find all extrema of the functionz=2x23y2+6xy+8x9y+3

22) A manufacturer has an order for 1000 units of wooden benches that can be produced at two locations. Letx1 andx2 be the numbers of units produces at the two locations. The cost function isC=0.25x12+10x10.1522+12x2+600 Find the number of that should be produced at each location to meet the order and minimize cost