Exam 3 review for Math 1190

Transcription

1 Exam 3 review for Math 9 Be sure to be familiar with the following : Extreme Value Theorem Optimization The antiderivative u-substitution as a method for finding antiderivatives Reimann sums (e.g. L 6 or R 8 ) The relationship between Reimann sums and definite integrals The relationship between definite integrals and area The Fundamental Theorem of Calculus Finding the average value of a function on [a, b] Applications of integration as covered in class Please ask if you think you see any errors in the questions below.

2 Sample Exam A:. Find the most general antiderivative of x 6.. Find e x dx. 3. Find 4. If 7 x 5x4 dx f (x)dx= x 3 +C and f() = 5, find f(x). 5. If f (x) = +3 and f(e) = 5, find f(x). x 6. Pick a u to use for u-substitution in e x3 x dx. 7. Explain how you could verify that 8. Use u-substitution to find of the result. x x +8 dx= ln(x +8)+C. 5x (5x 3 3) 7 dx= (5x3 3) 8 + C. Check this by taking the derivative 8 9. Find n and x if you were asked to find M 8 on the interval [,7].. If f(x) = x, find L 4 on the interval from [4,9].. What is the primary interpretation of the definite integral?. Use the facts that to evaluate the following: f(x)dx=, f(x)dx= 5, g(x)dx=, g(x) dx= 7. Use these integrals A) B) f(x)dx g(x) dx C) D) f(x)+g(x)dx 5f(x) 7g(x)dx 3. If f(x) is continuous and f(x) > on the interval [a,b], what does 4. Find x 5dx. b a f(x)dx = F(b) F(a) mean? 5. If f is a continuous function over a closed interval [a,b], then what is the formula for its average value, y av, over [a,b]?

3 4 6. Use the graph below to interpret f(x)dx in terms of the shaded areas A, B and C A y = f(x) C B 7. If a continuous function f has an antiderivative F over [a, b], then: b a f(x)dx = F(b) F(a) 3

4 Sample Exam B:. Find the most general antiderivative of x. 3. Find 7e x dx. 4. Explain the process you could use to find the national credit market debt in 9 based on the information below: For the years 5 through 9, the annual rate of change in the national credit market debt, in billions of dollars per year, could be modeled by the function: D (t) = 8.3t +7.3t+3648, where t is the number of years since 5. Note: D() = 4, Suppose P (x) = 3x 6x and P() = 9. Find P(x). 6. A fellow student wrote x +dx=x. Is that correct? If not, what did they do wrong? What should 7. Find they have done? Show the correct work. e 3x dx using any method. 8. Use u-substitution to find 9. Use u-substitution to find 3x 4x x 3 7x + dx. 3x 4x (x 3 7x +) 4 dx.. What are Reimann sums used for? Assume y = f(x) is a continuous, positive function. 4

5 . For g(x) = e x on the interval [,3] A) Find x if n = 6 B) Determine the x values what you would plug into g(x) if you were going to find R 4 C) Approximate the area between e x and the x-axis by finding R 4. Find an exact answer. 3. What is the primary interpretation of the definite integral? 4. What is the Fundamental Theorem of Calculus about? 5. Use the facts that 6. Find 7 f(x)dx= 3, to evaluate the following: A) f(x)dx= 8, f(x)dx B) x 5 +x 3 3x +9xdx. 7 7 g(x) dx=, g(x) f(x)dx C) g(x) dx= 3. Use these integrals f(x)+g(x)dx 7. Suppose F(x) is an antiderivative of f(x). Use the table below to answer the question that follows: x F(x) Find 7 f(x)dx. 8. If the units of y = f(x) are dollars and the units of x are days, A) what would the units of dy dx be? B) What about the units of b a f (x)dx? C) What about the units of the average value of f(x) over the interval [a,b] 5

6 Solutions to Sample Exam A:. x7 7 +C. e x +C 3. 7ln x x 5 +C 4. () 3 +C = 5 C = 3, so f(x) = x Finding the antiderivative gives that f(x) = ln x +3x+C. Solving for C gives ln(e)+3(e)+c = 5 C = 4 3e and f(x) = ln x +3x+4 3e 6. u = x 3 7. Either take the derivative of ln(x +8) or use u-substitution with u = x Letting u = 5x 3 3 gives du = 5x dx and dx = du 5x. Plugging stuff in, integrating and backsubstituting gives: 5x (5x 3 3) 7 dx 9. n = 8 and x = 7 = Here, x = 9 4 = 5 ( ) 4 4, so 5 L 4 = (f(4)+f(4+5/4)+f(4+(5/4))+f(4+3(5/4))) ( ) 4 5 = (+ /4+ 6/4+ 3/4). Since f is increasing, this would be an under estimate. 4. Area under the curve (so long as the function/curve is positive).. A) Since f(x)dx= f(x)dx= f(x)dx- f(x)dx+ f(x)dx f(x)dx=5-(-)=6 B) g(x) dx= g(x) dx+ g(x) dx=+7=9 C) D) 5f(x) 7g(x)dx=5 f(x)dx-7 g(x)dx= 5(5) 7(9) = One interpretation is the area between the curve f(x) and the x-axis. 4. ( (3)) (( )3 5( )) Note: 4 4 f(x)dx= A B. f(x)dx= B, 6 f(x)dx=a B +C, y av = b f(x)dx b a a 6 f(x)dx= B +C 6

7 7. The Fundamental Theorem of Calculus Solutions to Sample Exam B:. ln x +C 3. 7e x +C 4. First, find the antiderivative of D (t). Then use the initial condition to find the exact value of C. Then plug in t = 4, since 9 5 = The antiderivative of P (x) is P(x) = x 3 3x + C. Using the fact that P() = 9 we see that P(x) = x 3 3x + 6. They are wrong. They took the derivative, rather than the antiderivative. They should have taking the antiderivative. The correct answer is x3 3 +x+c 7. e3x 3 +C 8. Let u = x 3 7x +. Then du = 3x du 4xdx and dx = 3x 4x 3x 4x Substituting into the original gives u This simplifies to be 9. Let u = x 3 7x +. Then du = 3x 4xdx and dx = du 3x 4x u du=ln(u)+c = ln(x3 7x +)+C du 3x 4x 7

8 x 4x du Substituting into the original gives u 4 3x 4x This simplifies to be u du= u 4 du= u C = (x3 7x +) 3 3. Reimann sums are used to approximate the area under the curve.. A) x = b a n = 3 = 6 3 B) First, x =. Since R 4 must include the right-hand endpoint, 3 must be evaluated. Working back, we see that 3 / =.5,.5 / = and / =.5. So the four x-values are.5,,.5,3. C) x = /, so R 4 = (/)(e.5 )+(/)(e )+(/)(e.5 )+(/)(e 3 ) 3. Area under the curve (so long as the function/curve is positive). Be able to recognize this as a multiple choice answer. 4. The FTC gives a relationship between antiderivatives and the area question. That is, the ideas of rates of change (i.e. derivatives) are intimately linked to the area question! Cool indeed. 5. A) Since 9 7 C) 9 f(x)dx=8-3=5 B) f(x)dx= 7 7 f(x)dx+ 9 7 f(x)dx g(x) f(x) dx=--3=-4 6. = x6 6 + x4 4 x3 + 9x 7 = ( (7)6 6 + (7)4 4 (7)3 + 9(7) ) ( ()6 6 + ()4 4 ()3 + 9() ) 7. This is F(7) F() = 9 = 9 8. A) dollars day B) Dollars. Note, f (x) was being integrated. C) Dollars. Note the units of b units, converts the units back to dollars. a +C f(x)dx would be dollar days, but dividing by b a, with days as 8