MATH 2500: TAKE HOME 10 (30 points.) NAME: DUE: Wednesday, July 29th by 8 AM. DIRECTIONS: To receive full credit, make sure your work is neat and complete. Z 1. Evaluate the following integral using the substitution u = 2x 1: 1 2 4x dx (2x 1)3 HINT: Don't forget to change the limits on the integral from x's to u's. 2. Find the following indefinite integrals. Z cos(ln(x)) dx x Z e sin (x) dx 1 x2 Z e t e t dt e t + e t (a) (b) (c) 1 10 3. Find the following indefinite integrals. Z x 1 dx [ln(x)]2 + 4 Z 1 dx x 9x 2 25 (a) (b) 4. The rate of change of the amount in an investment account t years after money is invested is given by: A0 (t) = 250e 0.05t , Z (a) Find: 10 t0 A0 (t) dt. Find an exact answer as well as an approximation rounded to two decimal places. 0 (b) Interpret your answer to part (a) in terms of time and money in the account. (c) If $5000 was initially invested in the account how much money is in the account after 10 years? 10 Z 5. Consider 4p 1 + x 3 dx. 0 Write out each sum below then use a graphing utility to evaluate the sum, to five decimal places. (a) LS4 = (b) RS4 = (c) T4 = (d) S4 = 6. Earlier this semester, you used the Intermediate Value Theorem to prove is a solution to the equation cos(x) = x in [0, ] by showing f (x) = x cos(x) has a zero in the interval [0, ]. Use a graphing utility to implement Newton's Method to approximate the zero of f (x) = x cos(x) in the interval [0, ] starting with x0 = 0. Record the iterates below to five decimal places. x0 = 0 x1 = x2 = x3 = x4 = x5 = 10 BONUS: On the dashboard of pretty much every motor vehicle is a speedometer (which measures the speed of the vehicle) and an odometer (which measures the total distance traveled.) One of these instruments is connected with differentiation and one with integration. Match each instrument with the corresponding Calculus concept and in your own words, explain the connection. What big theorem relates the speedometer and odometer readings