MTH 210 Fall 2005 Test 1 Name: _Solutions_______________________
1. Find the following limits and answer the following questions.
A) 
Answer: 5/2 = 2.5 B)
Answer: 5/2 = 2.5
C) 
Answer: 5/2 = 2.5 D) f(0) = Answer: 5
E) Is f(x) continuous at x = 10? Answer: Yes
F) Is f(x)
continuous at x = 0? No . The limit as x
approaches 0 does not equal f(0).
G) 
Answer: 0
2. Find the following limits and answer the following questions.
A conservation organization releases 50 animals of an endangered species into a preserve. The population of the species is given by P(t). P is number of animals of the species in the preserve and t is time in years.
A) 
= Answer: 550
B) Write a sentence interpreting the meaning of the limit you found in A) to the context of the function.
Answer: The preserve environment will not support more than 550 animals
of this species in the long run.
C) The slope of the tangent line at t = 10 years is approximately 20.15. Write a sentence interpreting the meaning of the slope of the tangent line in the context of the problem.
Answer: The number of animals of this species is predicted to be
increasing by 20.15 animals per year 10 years after the original release.
3. Find the following limits and answer the following questions.
A) 
Answer: 108
B) f(6) = Answer: 108
C) Is f(x) continuous at x = 6? Answer: Yes . The limit as x approaches 6 equals f(6).
4. Find the following limits and answer the following questions.
A) 
Answer: positive infinity B) 
Answer: negative
infinity
C) 
Answer: Does not exist, since
the limit from the left does not equal the limit from the right.
5. f(x) = 0.03x2 + 254.50 0 < x < 100 y is exhaust temperature in degree Fahrenheit. x is the percentage load for a diesel engine . For example, x = 20 is 20%.
A) Find f(60) = Answer: 362.5
B) Find f ‘ (60)
= Answer: 0.06(60) + 0 = 3.6
C) Find the equation to the tangent line at x = 60.
Answer:
362.5 = 3.6(60) + b y = 3.6x + 146.5
146.5 = b
D) Write a sentence correctly interpreting the meaning of the derivative in the context of the meaning of this function.
Answer:
At 60% percentage load for this
diesel engine, the exhaust temperature is increasing by 3.6ºF per percent.
6. The position of an object with respect to a starting point is given by
s(t) = 10sin(0.2t) s(t) is in meters and t is in minutes
A) Find the s ‘(t)
at t = 
.
Answer:
s’(t) = 
s ‘ (5p/2) = 
B) 6) = Find
s ‘(t) at t
= 
Answer:
s ‘ (35p/6) =
C) Write a sentence interpreting the values you found in A) and B) with respect to the context of the meaning of this function.
Answer:
At t = 5p/2
minutes, the object is stopped for a moment as it turns around to travel back
toward the start. At t = 35p/6 minutes the object is heading back toward the
start at a velocity of 
meters/minute.
7. Use the formal definition to prove the value of the slope of the tangent line to function
y = ex at x = 4.
Answer:
8. Use the formal definition to prove the derivative of f(x) = cos(x).
Answer:
=
Use the short cut rules to find the derivative of the following functions.
9. f(x) = 
Answer:
f ‘ (x) = 
10. y = 
Answer:
y ‘ = ln(x)cos(x) + sin(x)/x
11. y = 
Answer:
y ‘ = 
12. f(x) = 
Answer:
f ‘ (x) = 