Math Calculus Final Name Be sure your solution is clear to the reader. Write in complete sentences.

Transcription

1 Math Calculus Final Name Be sure your solution is clear to the reader. Write in complete sentences. This portion of the test is to be done without any aid other than pencil or pen and eraser. As soon as you have finished, turn it in and pick up the second portion of the test. (On the second portion you can use your calculator and note card.) This first portion of the test is worth 80 out of 200 points. 1. A. Use the shortcut rules we have learned to find derivatives of the functions defined by the following expressions: Work below here if needed. a. 5t t b. π ln(2x) c. sin(x)cos(x) d. ln( x ) e x e. arcsin(3x) f. sec(x) B. Find the following integrals. Remember that a definite integral is a number and an indefinite integral is a family of antiderivatives. g. x 2 dx 2 1 h. 1 xdx i. csc 2 (x)dx

2 2. (5pt) a. Expand the binomial: (x+h) 6 Ans: (x+h) 6 = (5pt) b. What is the coefficient of x n-1 y in the expansion of (x+y) n? ans. 3. What is the period of the function q(x) = -2cos(2 x - π) - 2? 4. All parts of this question pertain to the function y = f (x) = ex 1 x 1. a. At what (if any) value(s) of x is the function zero? b. At what (if any) value(s) of x does the function have vertical asymptote(s)? c. At what (if any) value(s) of y does the function have horizontal asymptote(s)? d. Graph the function. The reader should be able to see clearly the values where the function crosses the x and y axes as well as the values of any asymptotes

3 Math Calculus Final Name Be sure your solution is clear to the reader. Write in complete sentences. On this portion of the test you may use your calculator and a 3 x5 note card. The questions below ask which of the functions listed here have various properties. List all the functions with the requested property. If no functions have the requested property write none. Values of the functions have been rounded off. s g(s) x h(x) t f(t) y k(y) Which of the functions listed above seem to be linear? 2. Which of the functions listed above seem to be exponential? 3. Which functions seem to have first derivatives that are always positive? 4. Which of the functions above appear to be concave up? 5. Using the table above for g(s), find an approximate value for g (0.1).

4 6. Suppose H(x) is a function where H'(x)=h(x) as given in the table above. a. What are likely value(s) for critical points of H? b. For each cirtical point listed in a., say whether it is a local maximum, minimum or neither. 7. Using the table above for g(s) find an approximate value for g(s)ds by finding LS, 0.3 the Left Hand Riemann sum. (Note the integral starts at 0.3, not 0.1) Pick ONE of the parts a or b below and do it. Only do ONE of them: Your choice. a. Use the definition of the derivative to show that d dx ( x ) = 1 2 x b. Find the smallest number of rectangles to use so that the difference between the LEFT 2 and RIGHT Rieman Sums that approximate e (x 2 ) dx is less than

5 10. Draw the graph of the derivative of the given function below the given function. a. b. c. d. Refer to the graphs given above to answer the questions that follow:

6 Give approximate ranges of x that satisfy the following conditions. If no range of values of x satisfy the condition, say none : e. For what values of x is the derivative of the given function in part b. negative? (6 pts.) f. For what values of x does the given function in part d. have a negative second derivative? (6 pts.) 11. Let F(t) be the number of inches of rain in the rain gauge t hours after midnight April 25 at SeaTac International airport. Let F'(t)=f(t). (6 pts.)a. What are the units of f? (6 pts.)b. What is the practical meaning of 4 the statement: f (t)dt = 0? (6 pts.) Find the average value of g(x)=x 3 between x = -2 and x = A 1000 square foot rectangular greenhouse is to be constructed. The south wall is to be glass windows which cost $100 per foot while the other 3 sides are standard walls which cost $50 per foot. a. (6 pts.) Draw a diagram of the rectangular greenhouse where the length of the south wall is labeled x. b. (6 pts.) Express the the total cost of constructing the 4 walls of the greenhouse as a function of x, the length of the south wall. c. ( 6 points EXTRA CREDIT ) Find the dimesions of the greenhouse that will minimize the cost of constructing the four walls. Give the dimensions to the nearest tenth of a foot.