Question Differentiate the following functions: f(x) = ln ((x^5(3x-5)^2)/((x+1)^12(1-2x)^7)   g(x) = (e^x-ln(x))^tanx   (a) \( f(x)=\ln \left(\frac{x^{5}(3 x-5)^{2}}{(x+1)^{12}(1-2 x)^{7}}\right) \) (b) \( g(x)=\left(e^{x}-\ln (x)\right)^{\tan x} \).

UCLAium-1919 is a radioactive element with a half-life of 140 days. Suppose that 10 mg of the element are placed in a lead container. Let y(t) represent the number of mg present t days later. Give exact answers only for all parts. No decimals, no rounding. The equation that represents the number of mg present t days later is y(t) = How many mg will be present after 10 weeks? Answer = In how many days will it take for 70% of the original sample to decay? Answer =

7–26. Lagrange multipliers Each function ƒ has an absolute maximum value and absolute minimum value subject to the given constraint. Use Lagrange multipliers to find these values. ƒ(x,y) = x2 subject to x2 + xy + y2 = 3

The Gourmand Cooking School runs short cooking courses at its small campus. Management has identified two cost drivers it uses in its budgeting and performance reports—the number of courses and the total number of students. For example, the school might run two courses in a month and have a total of 65 students enrolled in those two courses. Data concerning the company's cost formulas appear below: Instructor wages Classroom supplies Utilities Campus rent Insurance Administrative expenses Fixed Cost per Cost per Cost per Month Course Student $ 2,960 $280 $1,240 S 80 $4,600 $2,400 $3,600 44 S 5 $ For example, administrative expenses should be $3,600 per month plus $44 per course plus $5 per student. The company's sales should average $890 per student. The company planned to run four courses with a total of 65 students; however, it actually ran four courses with a total of only 57 students. The actual operating results for September appear below: Revenue Instructor wages Classroom supplies Utilities Campus rent Insurance Administrative expenses Actual $ 54,950 $11,120 $ 18,050 $ 1,970 $ 4,600 $ 2,540 $ 3,527 Required: Prepare a flexible budget performance report that shows both revenue and spending variances and activity variances for September. (Indicate the effect of each variance by selecting "F" for favorable, "U" for unfavorable, and "None" for no effect (i.e., zero variance). Input all amounts as positive values.) Required: Prepare a flexible budget performance report that shows both revenue and spending variances and activity variances for September. (Indicate the effect of each variance by selecting "F" for favorable, "U" for unfavorable, and "None" for no effect (i.e., zero variance). Input all amounts as positive values.) Gourmand Cooking School Flexible Budget Performance Report For the Month Ended September 30 Flexible Actual Results Budget Planning Budget Courses 4 Students 57 $ 54,950 Revenue Expenses: Instructor wages Classroom supplies Utilities Campus rent Insurance Administrative expenses Total expense Net operating income 11,120 18,050 1,970 4,600 2,540 3,527 41,807 13,143 $

13) The emf and the internal resistance of a battery are as shown in the figure. If a current of 8.3 A is drawn from the battery when a resistor R is connected across the terminals ab of the battery, what is the power dissipated by the resistor R? - 95.0 V 5.0 92 - - - - - A) 620 W B) 530 W C) 440 W D) 790 W E) 700 W

3. Use Lagrangian multiplier to maximize f(x, y) = x2 subject to x2 + xy + y2 = 3.

Find the linearization of y=sin(x) at x=π/4, and use your result to approximate sin(11π36). Enter exact values only and don't leave trigonometric functions unevaluated. (a) What is dxdx? (b) Using your linearization, approximate sin(11π36)sin(11π36). sin(11π36)