Problems appearing on your in-class final will be similar to those here but will have numbers and functions changed. For example, if problem 1 were selected for your in-class final exam, it might look like this:

1.   If f is a continuous function and , compute the exact value of . Simplify your answer as much as possible.

1.   If f is a continuous function and: , compute the exact value of . Simplify your answer as much as possible.

2.   Give an example of each of the following, or briefly explain why none exists:

a.   A function y = f(x) for which:
(i) .




(ii) .




b.   A function y = g(x) which is differentiable, but not continuous, at the domain point x = 2.





c.   A function y = h(x), continuous for all real numbers, which fails to have a derivative at precisely 3 of its domain values.

3.   Evaluate:

a.	b.
c.	d.
e.	f.
g.	h. Let , which arises in the study of the diffraction of light waves. Evaluate .

4. Find for each of the following:

1. 
   
   
   
2. 
   
   
   
3. 
   
   
   
4.  (Leave answer in terms of t.)
   
   
   
5. 

5. An offshore oil well is located in the ocean at a point W, which is 5 mi from the closest shore point A on a straight shoreline. The oil is to be piped to a shore point B that is 8 mi from A by piping it on a straight line under water from W to some shore point P between A and B and then on to B via a pipe along the shoreline. If the cost of laying pipe is $100,000 per mile under water and $75,000 per mile over land, where should the point P be located to minimize the cost of laying the pipe?

6.

1. Find both and in terms of x and y only from the equation .
   
   
2. Evaluate and when x = 2 and y = 1.     = ____________; = ____________

7.   Given the areas as shown for the graph of the continuous function , evaluate each of the following integrals.

8.

1. State the mean value theorem for a function on a closed interval and illustrate with a simple sketch.
   
   
2. Given show that there is no number c between a and b satisfying the conclusion of the mean value theorem.
   
   
3. Explain why the mean value theorem does not apply to the function given in part b.

9.   Determine whether the following statements are true or false. Briefly justify your answers.

1. If , then f(x) = g(x).
   
   
2. If , then f(x) = g(x).
   
   

10. Suppose that f(x) and g(x) are differentiable functions for such that for every x in the open interval a<x<b. Prove that f(b) - f(a) = g(b) - g(a).

11.   Two corridors 3 feet wide and 4 feet wide, respectively, meet at a right angle. Approximate the length of the longest non-bendable rod that can be carried horizontally around the corner, as shown in the sketch. Disregard the thickness of the rod. Round your answers to two decimal places.

12.   Let ; compute the following:

13.   Given that = f(u) + C, express each of the following integrals in terms of the function f.

14.   A particle is traveling upward and to the right along the curve . Its x-coordinate is increasing at the rate m/sec. At what rate is the y-coordinate changing at the point ?

15.   Suppose a shoreline has the shape of the parabola where x and y are measured in miles, and that a fog light located at (0,2) revolves at the rate of ½ radian per second. How fast does the x-coordinate of the point of illumination on the shoreline change at the instant the point (1,1/5) is illuminated?

16.   Sketch a graph of the curve y = g(x) from x = -5 to x = 5. Points (2,1) and (4,0) are on the graph of the function; the function has origin symmetry. You are also given that x = ±3 are asymptotes, that and that:

17.   A cylindrical tin boiler of given volume V₀ has a copper bottom and is open at the top. If sheet copper is 5 times as expensive as sheet tin per unit area, find the most economical dimensions (height and radius) for constructing the boiler.

18.   Suppose f, g, and h are differentiable at x = 3 and that . Find if y is as follows. Simplify your answer.

19.   Two moving particles have acceleration (at time t seconds) given by a₁ = 4t + 4 and . Assuming that both particles start from rest at t = 0, do they ever again have the same velocity? If so, when?

20.

1. Define what it means for a function f(x) to be continuous at x = a (where a is a real number).
   
   
2. Find the values of the constants a and b so that y = f(x) is continuous, and draw a sketch of its graph.
   

21.   Consider the function y = F(x) defined by .

1. Evaluate the following:
   
   
2. Find all maxima, minima, and inflection points of y = F(x). Over what interval is the graph of y = F(x) concave downward?
   
   
   
3. Use your calculator to approximate the values F(1), F(2), and F(3).
   
   
   
4. Use parts (a), (b), and (c) and any symmetry the function may possess to make a careful sketch of the curve y = F(x) on the interval: -3 ≤x ≤ 3.
   

22.   Solve for y in terms of x. if:

a.   (i) Any constant will do, for example: y=1. (ii) Any functions that concave down will do, for example: y = -x².

b.   Impossible: we know that at any point where is finite, the function must also be continuous.

c.   How about this function:

1. 2/9
2. 3
3. 2/5
4. 4
5. 
6. e^ab
7. 2
8. 

a.  

b.  

1. 21
2. 0.6
3. 0.8
4. -12/45
5. 0.6

1. Any function f(x) that is continuous on the closed interval and is differentiable for a<x<b must have at least one x-value c between a and b where
2. Since and It is impossible for , therefore there is no such c value.
3. Since is undefined at x=0.

a.   False
b.   True

10. By definition, f(x) is an antiderivative of and g(x) is an antiderivative of . Since =, g(x) is also an antiderivative of . According to part (a) of Exercise 53 in Sec. 4.9, two antiderivatives of the same function differ only by a constant. That is, f(x) − g(x) = K, or f(x) = g(x) + K, where K is a constant.
At x = a, and x = b, we have

11.  

12.

1. 
2. 
3. -6(2x - 5) ^-2

13.  

14.   m/sec

15.   0.963636 unit/sec

16.      

17.   ; and

18.

a.   37/4
b.   192+80ln2

19.   t=3/5 sec

20.

a.   f(x) is continuous at a domain number a if
b.   a = 3, b = -8

21.

a.   F(0)=0, F'(0)=1, F''(0)=0
b.   There is no maximum, and no minimum. F(x) is concave downward for all x>0.
c.  
d.  

22.  

Updated: October 24, 2006