Math 151 Week in Review
Sections 3.9-3.11
1. Find a unit normal vector to
a) the line parameterized by x  1  t , y  2  3 t .
b) the curve parameterized by x(t) = 2cos(t) y(t)= 3sin(t) at t = π/6.
2. Find the points on the curve where the tangent is horizontal or vertical.
a) x  t 3  3t 2
y  t  3t
3
b) x  a (cos t  cos 2 t )
(#12, Stewart section 3.9)
y  a (sin t  sin t cos t )
(#16, Stewart section 3.9)
Section 3.10 Related Rates
3. A snowball is melting so that its radius is decreasing by 2cm/min at the instant that the radius is 10cm.
How fast is the volume decreasing at this instant?
4. (Similar to Exam II, Sp'05 #18)
A helicopter is flying in a straight line at a constant height of 0.5 miles. Its speed is 120 mph.
The distance between the helicopter and a car on the road is 1 mile and is increasing by 30 mph.
How fast is the car going?
5. (Similar to Exam II, Fall'04 #17)
A camera is positioned 5000 feet from the base of a rocket. The rocket is fired vertically upward. At the
instant that its height is 2500 feet, its speed is 400 feet per second. How fast is the angle between the
camera and the rocket changing at this instant?
6. A 6 ft. tall man is walking away from a 10 ft tall light post at 2 ft./sec.
a) How fast is his shadow lengthening when he is 4 feet from the base of the post?
b) How fast is the distance between the light and the tip of his shadow increasing when he is 4 ft from
the base?
7. (#18 on pg 220 of Stewart) A trough is 10 ft long and its ends have the shape of isosceles triangles
that have a base of 3 ft. and height of 1 ft. The trough is being filled with water at the constant rate of
12 cubic feet per minute. How fast is the water level rising when the water is 6 inches deep?
3-11
8. The resistance of an elec tric wire of diameter D is given by R 
KL
D
2
where K and L are constants.
If the error in D is plus or minus 2%, find a) the approximate error in R and b) the approximate relative
error in R.
4
9. The volume of a sphere of radius r is V=  r 3 .
3
a) If the error in the radius is plus or minus 1%, what is the relative error in V?
b) If the radius is 2 with an error of plus or minus 0.02, what is the approximate error in V?
10. Use linear approximation to estimate
a)
3
25 . b)
4
17 .
  
.
20


11. Use quadratic approximation to estimate cos 
12. Given f ' ( x ) 
x  3 and f(1) = 5, approximate f(1.1) using
4
a) linear approximation.
b) quadratic approximation.
13. If P dollars is invested at annual interest rate, r , compounded continuously, then the accumulated
amount at the end of t years is given by A ( t )  Pe
a) Find the linear approximation to A(t) about t=0.
b) Find the relative rate of change of A(t).
rt
. Assume the interest rate is fixed.