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\fThe weekly sales (in thousands) of a new product after x weeks of advertising is S(x) = 5,000 - 1,700e .*. Find the rate of change of sales (in thousands per week) after the following. (Round your answer to the nearest integer.) (a) 1 week 154 thousand per week (b) 10 weeks 63 thousand per weekFind the indefinite integral. (Use C for the constant of integration. Remember to use absolute values where appropriate.) (45x2 + 3x -1 + 30e3X) dxFind the indefinite integral. (Use C for the constant of integration. Remember to use absolute values where appropriate.) [liaew-dex .. X X Find the indefinite integral. (Use C for the constant of integration.) (3 x - 3) dxFind the integral by a substitution or by integration by parts, as appropriate. [Use C for the constant of integration.) [X7 |n(4x) dx Use integration by parts to find the integral. (Hint: Take u = x2, dv = xe* , and use a substitution to find v from dv. Use C for the constant of integration.) x3 ex- dxA drug taken orally is absorbed into the bloodstream at the rate of ire0'\" milligrams per hour, where t is the number of hours since the drug was taken. Find the total amount of the drug absorbed (in mg) during the first 7 hours. (Round your answers to two decimal places.) mg absorbed amount absorption rate Hours (a) Solve without using a graphing calculator: mg (b) Verify your answer to part (a) using a graphing calculator. mg A factory is discharging pollution into a lake at the rate of r(t) tons per year given below, where t is the number of years that the factory has been in operation. Find the total amount of pollution discharged during the first 6 years of operation. (Round your answer to two decimal places.) t r(t) = to + 1Find the area between the curve y = x2 + 2 and the line y = 2x 3 [shown below) from X = U to x = 3. J' 15 square units Find the general solution of the differential equation or state that the differential equation is not separable. y4y' = 10X _'.-'= i/SlengK) | J Solve the differential equation and initial condition and verify that your answer satisfies both the differential equation and the initial condition. y4y' 6X5 || ._.. y(0) A medical examiner called to the scene of a murder will usually take the temperature of the body. A corpse cools at a rate proportional to the difference between its temperature and the temperature of the room. It y(t) is the temperature (in degrees Fahrenheit) of the body t hours after the murder, and if the room temperature is 60, then y satisfies v = *0 32(y , 60) , y(0) = 986 (body temperature initially 98.6\"). (a) Solve this differential equation and initial condition. r : 38.6603\" + 60 ./ (b) Use your answer to part (a) to estimate how long ago the murder took place if the temperature of the body when it was discovered was 80\". [Hint Find the value of L' that makes your solution equal 80\".] (Round your answer to two decimal places.) t-_, hr Suppose that you meet 25 new people each year, but each year you forget 50% of all of the people that you know. If y(t) is the total number of people who you remember after t years, then y satisfies the differential equation y' = 25 - 0.5y. (Do you see why?) Solve this differential equation subject to the condition y(0) = 0 (you knew no one at birth).Find the integral by whatever means are necessary (either substitution or tables). (Remember to use absolute values where appropriate. Use C for the constant of integration.) 25 2 fl/ 7X dx X Find the integral by using the integral table on the inside back cover. (Remember to use absolute values where appropriate. Use C for the constant of integration.) 1 zy 16 - z2 dzFind the integral. (Hint: Separate the integral into two integrals, using the fact that the numerator is a sum or difference, and find the two integrals by two different formulas. Remember to use absolute values where appropriate. Use C for the constant of integration.) x + 1 *V9+ x2 dxFind the size of the permanent endowment needed to generate an annual $28,000 forever at a continuous interest rate of 7%. In a psychology experiment, rats were placed in a Tmazej and the proportion of rats who required more than t seconds to reach the end was as shown below. 03 f CLOBeD'DSS o's r Use this formula to find the proportion of rats who required more than 10 seconds. (Round your answer to one decimal place.) ml . n. A straight kite string makes an angle of 40 with the ground, as shown in the diagram. (D If y = 210 feet of string has been let out, how high [in ft) is the kite? [Round your answer to one decimal place.) A boat sailing due West toward a pointy = 250 feet directly North of you makes an angle of 48, as shown in the diagram. Find the distance [in ft) between you and the boat. [Round your answer to the nearest integer.)