Students are allowed to use the textbook Single Variable Calculus: Early Transcendentals by James Stewart, notes, and a calculator. No other resources (friends, tutors, classmates, math reference websites, etc.) are allowed. All work must be shown to earn any credit. If techniques beyond the scope of Chapter 2 are used, no credit will be earned.
Sign the HONOR CODE at the end of the exam verifying no other resources were used.
Each problem is worth 5 points unless otherwise noted. There are 80 points possible.
1. Use the graph of ??(??) to complete the following parts. (10 points)
a)
b)
c)
d) lim ??(??) = _________ ???8
e) ??(2) = __________
f) ??(2) = _________
g) ??(3) = ____________
h) ??(3) = __________
i) State the intervals over which ?? is continuous._______________________________________
j) State the numbers at which ?? is not differentiable. ____________________________________
2. Consider the function ??(??) picture below. Write the equations of the asymptote(s) of the function, if any. Justify your answers using limits.
3. The cost (in dollars) of producing ?? units of a certain commodity is
??(??) = 4,280 + 13?? + 0.03??2
Find the average rate of change with respect to ?? when the production level is changed from ?? = 100 to ?? = 120. Include units with your answer.
4. Use a table of values to estimate the value of the limit. Draw the table you used. Full credit will be earned if x values appropriately approach 4 and results from the table support your given answer.
5. Consider the following function.
{
a. Determine the values of for which exists. Show your work justifying your answer.
b. If does not exist for a particular value of , state why not.
6. If and are continuous functions with and , find .
7. Find the limit , if it exists. Use Limit Laws if possible.
8. Find the limit , if it exists. Use Limit Laws if possible. ???8
9. Find the limit , if it exists. Use Limit Laws if possible.
. Find the following limits, if they exist. Show work to justify your answers. a. lim ??(??)
???1+
b.
c. lim ??(??)
???1
11. Use the Intermediate Value Theorem to show that there is a root of the equation in the given interval. Be sure to state why the IVT can be used. Full credit will be earned by using complete sentences and mathematically correct reasoning.
??5 - ??3 + 3?? - 5 = 0, (1,2)
12. Find an equation of the tangent line to the curve at the given point. . Use the definition of a derivative and show all work to earn full credit.
, use the definition of a derivative to find ??(??).
14. For the function ??(??) whose graph is given, arrange the following numbers in increasing order:
15. The graph of ??(??) is pictured below. Draw a plausible graph of ??(??) on the same axis.
BONUS Use the graph of the function to state the value of lim ??(??), if it exits. If the limit does not exist, ???0
explain why. 2 points
I pledge my honor that on this exam I have neither given nor received assistance not explicitly approved by the instructor and that I have seen no dishonest work.
_____________________________________________________ (Student signature)