1. Find each requested anti-derivative. a) x dx 4 b) 3 c) e6 x dx x 2 dx d) 5 x dx 2. Find the specific function f(x) which satisfies the following information: a) f (= x) 3x 2 8 x and b) f = ( x) and 6x 2 f( 2 ) = 3 f( 3 ) = 1 (Hint: use substitution) 3. Use the Method of Substitution to find the values of each Integral b) x e5 x a) 4 x 2 ( x3 1)6 dx 2 +7 dx 4. Solve the differential equation. Find the function satisfying the given information. a) dy = 4 x 6 and dx y (2) = 1 dy b) = e 2 x + 3 dx 1 1 2 2 ( x 3) 2 + 4= x x + 5 on the interval [0,12] 9 9 3 using n = 4 rectangles and Right-hand endpoints. (Hint: a sketch might help) 12pts 5. Estimate the Area under the curve of f ( x)= 6. Given the following information regarding f(x), answer the questions below. 10 4 7 1 10 1 14 , f ( x)dx = 6 3 , f ( x)dx = f ( x)dx = 4 a) 10 f ( x)dx 1 b) 2 f ( x)dx 4 7 c) f ( x)dx 4 7. Find the value of each Definite Integral, NO DECIMALS 16 a) 5 b) xdx 4 6 x 2 + 7)dx 3 9 10 3 c) dx x 2 (x 2 d) e dx 4x 1 x2 4 x x 4 dx 2 1 e) 8. Use the Substitution Method to evaluate the Definite Integrals below 2 a) 6 2 5 x( x 1) dx b) x 10 x dx 6 0 9. Find the following anti-derivatives. a) (10 x 4 + 2 )dx x5 3x xe dx 2 b) 10. Find the specific function f(x) which satisfies all of the following: 1 a) f ( x) = and f (11) 2 4x + 5 b) f ( x) = x3 4 x and f (6) = 4 11. Evaluate each of the Definite Integrals. NO DECIMAL ANSWERS. 2 2 a) (2 x 3 b) 36 x 2 ( x3 + 1)5 dx 3 x )dx 2 2 0 9 5 13 2 9 5 12. Given= f ( x)dx 10,= f ( x)dx 7,= f ( x)dx 16 , find 9 13 a) f ( x)dx b) 2 5 f ( x)dx 13 13. Find the Area bounded by = y x 2 and = y 2x + 8 3 14. Find the Area bounded by y x= = and y 9 x 15. Ed starts a 401k account at age 27, putting away $6000 each year. If the account earns 7.2% continuously, how much will be in the account when Ed reaches age: (i) 60 (ii) 65 (iii) 70 16. An investment opportunity is presented, which will provide a continuous income stream worth $75000 each year for the next 8 years. Assuming an earnings rate of 5% continuously, what is this worth in today's dollars? 17. Marie has built up a retirement account with a total value of $1,250,000 and wishes to convert it into an annuity that will pay her out over the next 25 years. Assuming an earnings rate of 6.4% continuously, how much will this provide her each year? (Hint: use the PV formula) 0 x 10 elsewhere 1/10 18. A PDF is given: f ( x) = 0 a) Show that this satisfies Principles I and II of a Probability Density Function b) Use Principle III to find P( 2 < X < 8) 3(1 x 2 ) 19. A PDF is given: f ( x) = 4 0 1 x 1 elsewhere a) Show that this satisfies Principles I and II of a Probability Density Function b) Use Principle III to find P( -1/2 < X < 1/4 ) 20. Find each Improper Integral's value, if is Converges, or show why it Diverges a) 3 2 x5 dx b) 1 3 dx 3 x c) 3 2 x dx d) 1 2e 1 t /2 dt