Calculus C: Extra Credit
You may choose up to __________ problems to complete for extra credit. Each problem is
worth a maximum of __________ points.
1. Definition of a Limit: Let f be a function defined on some open interval that contains the
number a , except possibly a itself. Then we say that the limit of f  x  as x approaches a
is L , and we write lim f  x   L if for every number   0 there is a number   0 such that
x a
f  x   L   whenever 0  x  a   .
a. Clearly explain the definition of a limit (as stated above).
b. Explain the difference between a limit, a left-hand limit, and a right-hand limit.
c. Sketch the graph of f  x   4 x  5 and use it to show the definition of a limit (as
stated above) graphically.
d. Use this definition of a limit to prove lim  4 x  5  7 .
x 3
2. Definition of a Derivative: The derivative of a function f at a number x , denoted by
f  x  h  f  x
.
f   x  , is given by f   x   lim
h 0
h
a. Clearly explain the definition of a derivative (as stated above). Be sure include
the meaning of x, f  x  , f  x  h  , h , and “limit” in your explanation.
b. Explain how this definition of a derivative relates to secant and tangent lines.
c. Sketch the graph of a function and show how it relates to the definition of a
derivative (as stated above).
d. Use this definition to prove that if f  x   sin x , then f   x   cos x .
3. Related Rates. For each of the following related rates problems, draw a clearly labeled
diagram, express all given quantities symbolically, express the equation that relates the given
rate(s) to the unknown rate(s), and show all work leading to your final answer.
a. At noon, ship A is 100 km west of ship B. Ship A is sailing south at 35 km/h and
ship B is sailing north at 25 km/h. How fast is the distance between the ships
changing at 4:00pm?
b. Gravel is being dumped from a conveyor belt at a rate of 30 fte/min, and its
coarseness is such that it forms a pile in the shape of a cone whose base diameter
and height are always equal. How fast is the height of the pile increasing when
the pile is 10 ft. high?
c. A runner sprints around a circular track of radius 100m at a constant speed of 7
m/s. The runner’s friend is standing at a distance 200m from the center of the
track. How fast is the distance between the friends changing when the distance
between them is 200 m?
d. Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the
radius of the oil spill increases at a constant rate of 1 m/s, how fast is the area of
the spill increasing when the radius is 30 m?
4. Definition of a Definite Integral: If f is a continuous function defined for a  x  b , we
ba
divide the interval  a, b into n subintervals of equal width x 
. We let
n
x0  a, x1 , x2 , , xn  b be the endpoints of these subintervals and we let x1, x2 , , xn  be
any sample points in these subintervals, so xi  lies in the ith subinterval  xi 1 , xi  . Then the
definite integral of f from a to b is

b
a
n
f  x  dx  lim  f  xi x .
n 
i 1
a. Clearly explain the definition of a definite integral (as stated above).
b. Sketch the graph of f  x   4 x and use it to show the definition of a definite
integral (as stated above) graphically. Hint: you may need more than one sketch.
c. Sketch the graph of f  x   x3  6 x and approximate
 f  x  dx
3
0
using a:
i. Riemann sum with left-endpoints
ii. Riemann sum with right-endpoints
iii. Riemann sum with midpoints
Which Riemann sum best approximates
 f  x  dx .
3
0
Justify your choice.
5. Interpreting Integrals: A definite integral is a number, while an indefinite integral is a
function or a family of functions.
a. Clearly explain the meaning of the symbols “  ”, “ f  x  ”, and “ dx ” in the
expression
 f  x  dx , both verbally and graphically.
b. Clearly explain the difference in meaning between a definite integral, an
indefinite integral, and an improper integral. Give an example of each.
c. Explain the difference in meaning between
d. Sketch and label the function f  x  
 f  x  dx and 


f  x  dx .
sin x
. Explain and illustrate the meaning of
2x
 f  x  dx , as it relates to your sketch.
5
0
e. Sketch and label the function f  x  



x 1
. Evaluate
x2
 f  x  dx ,  f  x  dx , and
5
0
f  x  dx . Show all work leading to your final answer.
6. More Proofs:
a. Using a Direct Proof, show that the sum of two rational numbers is a rational
number.
b. Using a Proof by Contradiction, show that if a is a rational number and b is an
irrational number, then a  b is an irrational number.
c. Using a Proof by Contrapositive, show that if x and y are two integers whose
product is even, then at least one of the two must be even.
d. Using a Proof by Mathematical Induction, show that for all positive integers n ,
n  n  1 2n  1
12  22  32   n2 
.
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