AP Calculus BC
Lesson 123 - Vector Valued Functions
Name:
If f is a function whose input is a real number and whose output is a vector, we call f a vector-valued
function.
Example: If x(t ) and y (t ) are real valued functions, then
f (t )  ( x(t ), y (t ))
is a vector valued function. In physics, they write f (t )  x(t )i  y (t ) j .
We call these parametric function.
Each coordinate functions are called component functions.
I've attached your homework to these notes. There are no need for practice problems.
The homework below is a PRACTICE TEST for next week. Do the work on a separate sheet of
paper and staple it to this worksheet. Please make sure it is legible. Write the answers next to the
problems below.
3
1) Find the arc length of the function f ( x)  2 x 2 from x = 0 to x = 3.
2) Let f (t )  3ti  cos( t ) j .
dy
.
dx
a) Find
b) Find
d2y
.
dx 2
3) Graph the polar curve r  3  2sin( ) .
θ - axis
4) For each of the following series, decide whether it converges or not. If it doesn't state why, if it does
find the value of the sum.

a)
3
5
k 2
n
n2  3
2
k  2 2n  n  2

b) 
6
k 3 n ( n  3)
Written by Bailey
3·2n  4n

7n
k 2


c) 
d)
AP Calculus BC
Lesson 123 - Vector Valued Functions
Name:
5) Use partial fractions to solve the integral
 2 4 x  37 dx
x  11x  28
6) Use trigonometric substitution to solve the integral

36  x2 dx
7) Use trigonometric identities to solve the integral
 sin ( x)dx
5
8) Use integration by parts to solve the integral

7 x3 cos(3x)dx
9) Let R be the region between f ( x)   36  x 2 and g ( x)  36  x 2 . Find the volume
of the solid with cross-sections perpendicular to R and the x-axis are squares.
10) Let R be the region bounded by the graphs of f ( x)  e x and g ( x)  1  x 2 , x = 0 and
x = 1.
a) Find the area of R.
b) Find the volume of the solid generated by revolving R around the x-axis.
(Washer Method)
c) Find the volume of the solid generated by revolving R around the y -axis.
(Cylindrical Shells)
11) The length of a solid cylindrical cord of elastic material is 32 inches. A circular cross
section of the cord has a constant radius of 0.5 inches. Suppose the cord is stretched
lengthwise at a constant rate of 18 inches per minute, and it maintains its cylindrical
shape and constant volume. At what rate is the radius of the cord changing?
12) Let y  cos( y )  x  1 .
a) Find the equation for the tangent line at the origin.
b) Is the graph of the curve defined by the equation concave up, concave down, or neither
at the origin. Justify your answer.
Written by Bailey