Calculus I Final Exam. April 2003. Name Math-
ematically justify your answers (show work). Non-graphing, non-calculus cal-
culators allowed. Simplify and complete all computations as much as possible.
Circle answers. Remember that sloppiness (missing parentheses, illegible work,
etc.) will cause lower scores.
PART I. Shorter problems (3 points each).
1. Find g0(x) if
(A) g(x) = 2x3+x−2
(B) g(x) = sin(5x2+ 4)
(C) g(x) = x3cos(x)
(D) g(x) = x
1+x2
(E) g(x) = 5x
(F) g(x) = sin2(x+x3)
2. Find lim
x→5
3x2−75
x−5
3. Find R2
1x3dx
4. Find R3
xdx
5. Find an equation of the line tangent to the curve y=1
xat the point
(2,1
2).
PART II. Longer problems (10 points each).
6. Let g(x) = 1
x.Use the limit definition of derivative to show that g0(a) =
−1
a2.
7. Suppose a function yis defined implicitly by equation (*) y2−5xy −x3=
13
(A) Find y0=dy
dx .
(B) Find the equation of the line tangent to the graph of equation (*) at the
point (1,−2).
8. Let g(x) = x4−4x3,Using calculus tools, identify all significant features
of the graph of g, and sketch the graph.
9. A cylindrical container with volume 1000cm3is to be made. Find the
radius and height that will make the total surface area minimal.
10. A tank has the shape of a cone with height 10 meters and radius 3
meters at the top. Water is pumped into the tank at the rate of 2 cubic meters
per minute. How fast is the water rising when the water is 4 meters deep?
11. Find the largest and smallest values (and where they occur) for f(x) =
x3+3
xon the closed interval [1
2,2].
12. Evaluate the following:
(A) R9
1
1
√xdx
(B) R2
0(x3+ 1)2dx
Surprise Extra Credit (10 points)
Use Riemann sums and the limit definition of the integral to find R3
0x2dx.
You may need the formulas
N
P
k=1
k=N(N+1)
2,
N
P
k=1
k2=N(N+1)(2N+1)
6.
1
