Solutions to the MAT220 Spring 2011 Practice Final Exam

Solutions to the MAT220 Spring 2011 Practice Final Exam.

Use the above graph to answer questions #1 and #2

1. A) 1B) 1C) 1D) 1

E) F) G) H) DNE

I) DNEJ) 1K) 1L) 1

M) N) O) DNE

2. A) What are the zeros for ?

B) Is a function? Explain. Yes. You can see the graph passes the vertical line test!

C) Does have an inverse function? No. The inverse will not be a function because is not 1-to-1

D) For what value(s) of x if discontinuous ?

E) For which value(s) of x is the discontinuity removable? 3

F) What is

3. Find WITHOUT using L’Hopital’s rule!

4. Find WITHOUT using L’Hopital’s rule!

5. . Find by using the product rule. Write your answer as a polynomial with terms in descending order.

6. . Find by using the quotient rule. Write your answer as a simplified rational expression with the numerator and denominator factored.

7. by implicit differentiation. If you’ve simplified your answer properly, you should have a single term in the numerator and a single term in the denominator.

8.

A) Is continuous at x = 0? Explain. Is continuous at x = 0? Explain. Is continuous at x = 0? Explain.

is continuous at x = 0 because each “piece” = 2 when you substitute in x = 0.

is NOT continuous at x = 0 because has a vertical asymptote at x = 0.

is continuous at x = 0 because each “piece” = 0 when you substitute in x = 0.

B) Is differentiable at x = 0? Explain. Is differentiable at x = 0? Explain. Is differentiable at x = 0? Explain.

is NOT differentiable at x = 0 because and the two “pieces” don’t match at x = 0!

is NOT differentiable at x = 0 because has a vertical asymptote at x = 0.

is differentiable at x = 0 because and each “piece” does match at x = 0!

C) What is the absolute max of ? What is the absolute max of ? What is the absolute max of ?

The absolute max of is 2 (Draw the function!)

The absolute max of DOES NOT EXIST. Draw the graph to see that as

The absolute max of is 1 (Draw the graph…..remember what sine and cosine look like)

D) Does have real zeros? Does have real zeros? Does have real zeros?

All three function have real zeros. has two real zeros, has one real zero and has an infinite number of real zeros. Just “look” at how many times each graph “crosses” the x-axis.

F) Does have an inverse function? Does have an inverse function? Does have an inverse function?

None of the three functions have an inverse that is a function. When you view the graph of each you will see that NONE of them pass the horizontal line test thus none of them are 1 – to – 1 functions and consequently their inverses will NOT be functions.

9. Find the absolute max and absolute min of AND state where (x-values) they occur.

10. Find any value of “c” guaranteed by the Mean Value Theorem for .

11. The position function of an object moving along the x-axis is given by

A) During what time intervals is this object moving to the left? Moving to the right?

From my table above we can see that the object is moving left during the time interval

and the object is moving right during the time intervals and

B) During what time intervals is this object speeding up and slowing down?

From my table above we can see that the object is speeding up during the time intervals and

and the object is slowing down during the time intervals and

C) What is the total distance travelled by this object after 3 seconds (round your final answer to two decimal places)?

12. Find USING L’Hopital’s rule! (How does this answer compare to what YOU got on #3?)

Note: L’Hopital’s rule applies since this limit is of the form

(YES, I got the exact same answer as then I did this same exact problem a different way in #3)

13. Find USING L’Hopital’s rule!(How does this answer compare to what YOU got on #4?)

Note: L’Hopital’s rule applies since this limit is of the form

(YES, I got the exact same answer as then I did this same exact problem a different way in #4)

14. . Find all relative max’s and min’s. Round values to two decimal places.

15. Find all P.O.I for the function in problem 14 above.

16. Evaluate

17. Evaluate . Give an exact answer!

18. A) if is an odd function with . SHOW WORK!!!!

B) if is an even function with . SHOW WORK!!!!

19. Evaluate by sketching the region whose signed area is represented by this integral and using an appropriate formula from geometry.

20. Evaluate . Give an exact answer.

21. Evaluate

22. Find

23. Evaluate . Be sure to simplify your answer as much as possible!!!

24. Evaluate by rewriting it as the sum of two integrals and evaluating using the FTC

(How does this answer compare to what YOU got on #19?)

Yes this is the exact same answer that I got when I did this exact same problem on #19 using geometric formulas.

25. Let For what value(s) of x will ?