M152: Calculus II Midterm Exam Review

Transcription

1 M52: Calculus II Midterm Exam Review Chapter : Mean Value Theorem. - Know the statement and idea of Mean Value Theorem. - Know how to find values of c making the theorem true. - Realize the importance of the Mean Value Theorem for the Fundamental Theorem of Calculus. - Know that the only functions with f (x) = are constant functions. 4.9 : Antiderivatives. - Know how to find basic antiderivatives (including guessing). - Understand how to use an initial condition to find a particular antiderivative (instead of a general family). - Know all the basic rules for finding antiderivatives. Chapter : Areas. - Understand how to find approximate areas under curves given by data and by functions. - Know how to use rectangles to approximate areas under curves using right (or left) endpoints. - Understand summation notation and how to use it. - Recognize the role the limit plays in finding areas. 5.2 : The Definite Integral. - Understand the notation for the definite integral: b a f(x). - Know the definition of the definite integral using sums (same as in 5.). - Be able to find the definite integral of basic functions using the definition. - Understand how to manipulate sums to help evaluate a limit (pg. 38). - Know how to find general points x i as well as x for given problems. - Know how to find the definite integral by interpreting the area geometrically. - Know the properties of the definite integral (pg ). - Understand what the definite integral computes ( signed area ). 5.3 : Fundamental Theorem. - Know the statement of the Fundamental Theorem of Calculus and why it s important. - Understand how to interpret the function g(x) = x a f(t)dt. - Be able to apply the FTC to find derivatives of functions like the one above. Understand the role of the chain rule. - Understand the inverse process between derivatives and integrals. - Know that there are situations where the FTC cannot be used. - Be able to use the FTC to evaluate definite integrals using antiderivatives. 5.4 : Indefinite Integrals. - Understand the notation for the indefinite integral: f(x). - Understand the difference between b a f(x) and f(x). - Be able to clearly describe the two notations given above. - Know all of the basic indefinite integrals and rules for indefinite integrals (pg. 43). - Understand that a definite integral measures net change. - Be able to use definite integrals to measure total distance traveled as well as displacement. - Be comfortable with interpreting information from graphs to figure out different areas.

2 2 5.5 : Substitution. - Understand how to perform substitutions to evaluate indefinite and definite integrals. - Be able to try different substitutitions to find the right answer. - Understand that when substituting you must make sure every part of the integral gets changed to the new variable. - Understand that substitution is changing your frame of reference and that there are many consequences of your choice for a substitution. - Be comfortable working both symbolically and with real functions. - Know how to interpret definite integrals for even and odd functions. Chapter : Area Between Curves. - Know how to find the area between two curves by interpreting the area as an indefinite integral. - Understand how the area depends on the situation of the problem (which curve is on top) and that you may have to break problems up in to pieces to solve them. - Be comfortable setting up area problems as definite integrals and then solving these integrals. - Understand the difference of using y as a independent variable instead of x in order to make some problems easier. - Know how to find intersection points of curves to break problems into pieces. 6.2 & 6.3 : Volumes. - Understand how to interpret distances using the x-y coordinate system. - Be comfortable working with small cross-sections (i.e. differential elements) and understanding how to produce geometric information about these elements. - Understand the different shapes that these differential elements form depending how you rotate them. - Know how to find areas of the different shapes produced by the differential elements. - Know how to find the volume of a differential element depending on a particular problem. - Understand the role the definite integral plays in finding the volume of a solid. - Be comfortable with drawing graphs of functions and understanding intersection points as well as what rotation about an axis means. - Understand how to think about cross-sections (as above) and determine information. - Be comfortable working with x or y as your variable (depending on different situations). Chapter : Integration by Parts. - Know how integration by parts works and where it comes from. - Understand the integration by parts formula and how to use it. - Be comfortable using integration by parts multiple times and looking for patterns. - Be comfortable using integration by parts to find definite integrals. 7.2 : Trigonometry Identities. - Be comfortable with the major trigonometry identities (they will be supplied for you, but be comfortable with them). - Be able to find integrals of the form cos m (x) sin n (x) using identities and substitution. - Be able to find integrals of the form sec m (x) tan n (x) using identities and substitution. - Be comfortable with the basic integrals for sin(x), cos(x), sec(x), and csc(x). - Understand how to use the double angle identities (e.g. sin 2 (x) = 2 2 cos(2x)). 7.3 : Substitution with Trigonometric Functions. - Know how to recognize expressions that look like the three standard trigonometry identities. - Be comfortable substituting trigonometric functions (sin(θ), sec(θ) or tan(θ)) for x. - Know how the trig functions relate to a right triangle and be able to get necessary information from these triangles.

3 Review Problems. Please note that these are examples of problems. Do not expect to see exactly the same problems on the exam with different numbers. I expect some amount of problem solving on the exam. Also, do not expect the exam to be anywhere near as long as this set. Before Quiz. () What are the antiderivatives of the following basic functions (find the general form): 3 (a) f(x) = cos(x) (b) g(x) = x 2/3 (c) h(x) = 4x 7 + x (d) f(x) = sec 2 (x). (e) g(x) = x 2 (f) h(x) = e x sin(x) sec(x) tan(x) (2) Suppose that F (x) = 32x 3 + e x. Find F (x) satisfying F () = 5. (3) When we estimate distances from velocity data (as the area under the graph of velocity), it is sometimes necessary to use times t, t,... that are not equally spaced. Use the following data and right endpoints to estimate the distance above earth of the space shuttle Endeavour 62 seconds after liftoff. Time Velocity Hint: remember you are using rectangles, but the bases may be different for each rectangle. (4) Using the definition, write the expression for 8 (5) Use the definition of the definite integral to compute (6) Based on your previous answer, what is 2 (7) Using the definition of the definite integral, compute negative? (8) Explain what is meant by of Calculus. 6 (9) Draw a graph of f(x) = x 2. Determine () Let h(x) = () Compute d (2) Suppose that (3) Suppose that x x 2 ln(x) (you do not need to compute this). 2 (x 2 + ). (x 2 + )? How do you know? 2 (x 4). Can you explain why your answer is (x + 3). Compute this without using limits or the Fundamental Theorem sin(t) cos(t)dt. Find h (x). [ x (4u ln(u) + e 4u ]. u 3 )du b a 7 4 g(x) = and f(x) = 2 and c a 4 (4) Explain in your own words what is meant by f(x). g(x) = 3. Determine g(x) =. Determine b a q(x). c b 7 4 g(x). (f(x) g(x)).

4 4 (5) Can you use the Fundamental Theorem of Calculus to compute 5 5 x? (6) Explain how you would use the Fundamental Theorem of Calculus to compute these instructions? Why or why not? (7) Explain the importance of the Fundamental Theorem of Calculus. (8) Describe the function g(x) = x (9) In your own words describe the notation t 2 dt. You may use a picture if necessary. x 7 and what it means. 9 ln(x). Can you follow (2) Find the following integrals using any method you want: x x (a) (i) x (b) 5x (j) (c) (sin 2 (x) + cos 2 (x)) (k) (d) cot(x) (l) tan(x) sec 99 (x) 8 5 (x + 4x cos(x 2 )) x 3 x sin(x). cos(5x + 3) (e) (f) (g) (h) e t + 5 dt 8 5 e t (x 2 + 3x 2 ln(x)) cos(x 3 ln(x)) e 8x (x + ) + 2x + x 2 (m) (n) (o) (p) x 2 x 6 sec(θ) tan(θ) π + sec(θ) dθ π cos(x)f(sin(x)) 6 4x (2) Use algebra, trigonometry and substitution to explain why (22) Can we find (23) Find (24) Find e x dt. e π. (25) Suppose that Post Quiz. 4 e x2 using the tools we have? Explain how or why not. x 3 e x = 862. Find 6 3xe x. () Find the area bounded by the curves x = y 2 2, y =, y = and x = e y. (2) Find the area enclosed by the curves y = x 3 x + 6 and y = 3x + 6. π 2xe x 2 represents the same area as e sin(x) sin(2x)

5 (3) Use definite integrals to find the area bound by the curves y = x, y = 2 2x and y =. Check your answer using geometry. (4) The following is a graph of f(x) = x 5 (a) Suppose we take the region bounded by y = x, x = and x = 6 and rotate the region around the x-axis. Describe this by drawing a picture. (b) Draw a picture of a differential element being rotated the same way. (c) Find the volume of the solid drawn above. (d) Repeat the process, but revolve the region around the y-axis this time. (e) What is different between the two previous processes? (5) Below is a graph of the functions y = 6x, y = 3x and y = 3. (a) Find the volume of the solid when rotating around the y-axis. (b) Find the volume of the solid when rotating around the x-axis. (6) Find the volume of the solid generated by rotating the region bounded by y = x 2, y = and x = around the line x = 2. (7) Plot f(x) = cos(x) over the interval [ π 2, 3π ] 2. Find the volume of the solid obtained by rotating this region about the x-axis. Do the same for rotating about the y-axis. (8) Find the following indefinite integrals: (a) x sin(πx). (b) e x sin(x). (c) sin 5 (x) cos(x). (d) sin 2 (x) cos 2 (x). (e) sin 4 (x) cos 5 (x). (f) x x 2 8 (g) x 2 +4 x (h) 6 x 2

6 6 (9) Find the following definite integrals: (a) e x2 ln(x). (b) ex sin(x). (c) π sin2 (t) cos 2 (t)dt. (d) 5 (e) 2 5x 2 x 2 +4 x (f) x 2. () Draw the area for the region described in part f of the previous problem. Do you know a geometry formula for this area or not? () Suppose that 5 (2) Find 2 2t (t 3) 2 dt. x = M. Find 4x+ in terms of M. (3) We know that t t dt = 2 t + ln( t ) ln( t + ) + C. Find 4 + e x. (4) Suppose that ex sec 2 (x) = (according to a computer). Find ex tan(x). (5) Find π/3 π/4 ln(tan(x)) sin(x) cos(x). (6) For the following integrals, say which integration technique you would employ (or try) first. (a) tan (x) x 2. (b) e x + e 2x (c) (π 7 + ) (d) x 2 +x 2 (e) x 2 +x 2 (f) sin 3 (x) cos 5 (x). (g) x 3 cos(x). (h) 3 ( t 2 ) 3 2 dt (7) If we use integration by parts with u = x to evaluate x sec(x) tan(x), what integral are we left with determining instead?