(a) Using calculus, find the area bounded by the two parabolas P₁(x)=x²-x + 1/2 and P₂(x) = x² + x + 1/2. (b) Estimate the area as a Type 1 Monte Carlo simulation, by finding the average value of P2 (x) - P₁ (x) on [0, 1]. Find estimates for n = 10 for 2 ≤ i ≤ 6. (c) Same as (b), but estimate as a Type 2 Monte Carlo problem: Find the proportion of points in the square [0, 1] x [0, 1] that lie between the parabolas. Compare the efficiency of the two Monte Carlo approaches.

A poster consists of a printed area and margins (See image). The area of the entire poster is A = 96 square inches. The goal is to find the dimensions x and y that maximizes the printed area. A = xy 3" A = A(x) = The area of the printed part, as a function of x and y is Now, A'(x) = x = Use the substitution principle to rewrite the area as a function of x. Hint: You are given the area of the total poster. 6 96 x+6 12 2" - 4 X 2" To determine optimal values, we need to look at A'(x) = 0. Solving for x and, ultimately for y, yields ✓inches Xinches y X 3" Question Help: Message instruct What is the maximum area of the printed part? A = 72 X square inches

Let f(x)=(225-x^2)^1/2. What is the linear approximate for f(12.1) to two decimal places?   note to expert: Please explain your steps. I am really struggling

A creep test is done at room temperature on a material on a metal alloy wire comprising of 30% Tin, 25% lead and a solid core copper of 40%. The test measures the increase in strain (Pa) over time (hr) while a constant load is applied to the test specimen wire. Fit the following data using quadratic regression and determine the strain for the duration time t = 5.625 hrs. (note use the rounded off coefficient values in calculating the function). (Answers should be in 3 decimal places) f(x) x2 x3 ху 0.452 Blank 1 Blank 2 Blank 3 Blank 4 Blank 5 1 0.724 Blank 6 Blank 7 Blank 8 Blank 9 Blank 10 0.918 Blank 11 Blank 12 Blank 13 Blank 14 Blank 15 4 0.966 Blank 16 Blank 17 Blank 18 Blank 19 Blank 20 0.948 Blank 21 Blank 22 Blank 23 Blank 24 Blank 25 5 Sum x Sum y Sum x? Sum x Sum x* Sum xy Sum xy Blank 26 Blank 27 Blank 28 Blank 29 Blank 30 Blank 31 Blank 32 Ao = Blank 33; A = Blank 34; A2 = Blank 35; f(5.625) = Blank 36 3.

Please answer only the first three square blanks

Differentiate: f(x) = 4x³ - (5/x²) O a. f(x) = 12x² + (10/x³) O b. f(x) = 12x² + 10x³ O c. f(x) = 12x² - 10x3 O d. f(x) = 12x2- (10/x³) If y = x*/v(1-2x), find y'. O a. (4x³ + 7x*)/(1 - 2x)'/2 O b. (4x3 + 7x4)/(1 - 2x)² O c. (4x3 - 7x4)/(1 - 2x)² O d. (4x³ - 7x^)/(1 - 2x)9/2

If f(2)=9 and f'(2)=2, estimate f(1.9).