Part 1: Integration problems from exams

Transcription

1 . Find each of the following. ( (a) 4t 4 t + t + (a ) (b ) Part : Integration problems from 4-5 eams ) ( sec tan sin + + e e ). (a) Let f() = e. On the graph of f pictured below, draw the approimating rectangles that are used to estimate the area under the curve between = and = according to the the Left Endpoint Rule; use n = 4 rectangles. Write an epression involving only numbers (including e) that represents the area estimate using these rectangles. Is your quantity an underestimate or an overestimate of the actual area? (c) Write a mathematical statement that epresses the area under the curve from = to = as a limit, again using the Left Endpoint Rule. Eplain any notation you use. (You should not evaluate this limit.) (d) Calculate the value of the definite integral you can.. Evaluate the following integrals, showing all of your work. (a) (c) (ln ) e, simplifying your answer as much as

2 4. A particle moves along a line, with acceleration (in meters/sec ) as a function of time t (in seconds) given by a(t) = t 7. Furthermore, at time t = second, the particle s velocity is 4 meters per second. (a) Find the particle s velocity function v(t). What is the particle s net change in position (i.e., its displacement) between the times t = and t = 7? (c) Find the total distance traveled by the particle between t = and t = For this problem, we define g to be the function + g() = f(t), where f() = + 5 Below is the graph of the function f: 4 f (a) Find g ( ). On what intervals is the graph of g concave down? (c) Determine the absolute maimum and minimum values of g on the interval [, 5]. (d) Find a formula that gives the value of g() for any 5. (Notice that this is just a small portion of the domain graphed above.) 6. The growth rate, measured in dollars per year, of a certain prestigious university s tuition can be modeled by the function h(t), where t is the number of years since 99: (a) Calculate 8 4 h(t) = (.5) t h(t). (You do not need to epress your answer in simplest form.) What are the units of your answer to part (a)? Write a sentence epressing the meaning of the quantity you found. (c) Find d 5 (.5) t, showing all steps in your calculation.

3 7. (a) Determine d arctan t. Find a function f and a value of the constant a such that 7. Let g be a function for which the following things are true: g() =, the graph of g, the derivative of g, is shown below: a f(t) = g (a) Apply the Evaluation Theorem to write a simple mathematical statement involving the definite integral b a g (t). Use a = and b = in your answer to part (a) to obtain an epression for g() in terms of an integral. (c) Use part and the graph above to fill in the table of values below for g(). In the space below the table, give a sentence (or more if necessary) eplaining your reasoning. (You may also use the net page if needed.) g(). Find each of the following. ( (a) csc t + t ) t Part : Integration problems from 5-6 eams ( 5( ) + ( )( ) e )

4 . (a) Let f() =. On the aes pictured below, draw the approimating rectangles that are used to estimate the area under the curve between = and = according to the Right Endpoint Rule; use n = 6 rectangles..5 Write an epression involving only numbers that represents the area estimate using these rectangles. (You do not have to simplify the epression.) Is your quantity an underestimate or an overestimate of the actual area? (c) Write a mathematical statement that epresses the area under the curve from = to = as a limit, again using the Right Endpoint Rule. Eplain any notation you use. (You should not evaluate this limit.) (d) Find the value of the definite integral. Justify your answer. (Hint: it s wise to avoid using the Evaluation Theorem.) 4. Each of the following epressions involves an integral. When the epression is evaluated, the result is either a number or a symbolic quantity involving one or more variables. For each, say whether the epression can be evaluated to: (I) a number, involving no variables, (II) a quantity involving the variable but not involving t, (III) a quantity involving the variable t but not involving, or (IV) a quantity that must involve both variables t, (i.e., none of (I)-(III)). No justification is necessary, and there is no penalty for guessing, but each line should have only one response. (Note: you needn t actually evaluate all of these epressions.) Epression I, II, III or IV Epression I, II, III or IV t t + t + d d t + t +

5 4. Let f() = t. (a) Find f( ) and f ( ). Find the intervals on which f is increasing or decreasing. (c) Find the intervals of concavity of f. (d) Using the information you found above, sketch a plausible graph of f. Be sure to label your aes as appropriate. 5. Evaluate the following integrals, showing all of your work. (a) 4 + ln / (c) (a ) (b ) t e t 5 ln 7. Suppose functions f and g are given by if <, f() = if, if >, and g() = f(t). (a) Sketch a graph of f; be sure to label the scale on your aes. Evaluate g( ), g(), and g(), showing your reasoning. (c) What is g (4)? Give complete reasoning. (d) Find an eplicit formula (i.e., no integral signs) that gives the value of g() in terms of, where lies in the interval. Justify your answer.