4. y"+y=t-u(t-1)t y(0)=0 y'(0) = 0
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help with number 4 show the steps to get the correct answer, the correct answer for that is (t-sint) - u(t-1)((t-1)-sint(t-1))
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- 3. Suppose that the temperature on a metal plate is given by the function T with T(x, y) = 100 - (x² +4y²) where the temperature is measured in degrees Fahrenheit and a and y are each measured in feet. Now suppose that an ant is walking on the metal plate in such a way that it walks in a straight line from the point (1, 4) to the point (5, 6). (a) Find parametric equations for the ant's a and y coordinates as it walks the line from (1,4) to (5,6) in one minute at a constant speed. (b) What can you say about dr dt and dy dt for every value of t? (c) Determine the instantaneous rate of change in temperature with respect to t that the ant is experi- encing at the moment it is halfway from (1, 4) to (5,6). Include units in your answer.The equation of a curve is y =x'+ar + bx, where a andb are constants. (1) In the case where the curve has no stationary point, show that a < 3h. (II) In the case where a=-6 and b = 9, fìnd the set of values of x for which y is a decreasing function of x.Suppose that the cost function C for a chemical lab to produce x liters sanitizer liquid per week is given by C(x)=5000+ Vx-0.001x°(RM). The lab set the price pand amount x liters sold follow the demand equation p(x)= 60–0.0005x° (RM). As an intern in the lab, you are required to find the break-even point, that is, how much it should produce the liquid per week in order to have neither a profit nor a loss. Use the Secant method with error tolerance 5x10² to approximate the break-even point. Begin your search with po-50 and pi=200. Show all the necessary working steps and write your answer in 4 decimal places.
- Find y' implicitly. 1. (x + y)² = 4y 2. (x + 2y)2 = x + 10 1 of 1Find General Solution of y" + y" = 0.(3) The demand curve of a firm is p 1200-21q and its total cost is C(q) = 24' - 66q° + 600q +1000 where q is the output of the firm (in thousands). (i) Derive an expression R(q), for the firm' s revenue function. (ii) Derive an expression II(9) for the firm' s profit function.
- Consider the function f(x)=(7-³) ¹0. Determine each of the following. 1. f(2)= 2. f'(2)= 3. Substitute the values of f(2) and f'(2) into the equation y f(2)= f'(2)(x − 2). (x - 2). y - 4. Solve the resulting equation for y. Type your final equation here . Your equation must be in the form y = mx + b.(3) The demand curve of a firm is p=1200 – 2 lq and its total cost is C(q) = 2q° – 66q² +600g +1000 where q is the output of the firm (in thousands). (i) Derive an expression R(q), for the firm' s revenue function. (ii) Derive an expression II(g) for the firm' s profit function. (iii) Is the rate of change of profit increasing or decreasing when the output level of the firm is 10,000 units? (iv) Determine the level of output at which profit is maximized.b) A particle moves along x- axis so that its position at any time t20 is given by the function s(t) =t³ – 12t +1, wh 's' is measured in feet and 't' is measured in seconds. (i) At what time does the particle change directions? (ii) (iii) When is the particle moving left? When is the particle slowing down?
- The procedure to find the stationary points of a particular function f(x, y) leads to the eq 3x² +18x3y² +27= 0, -6xy +by = 0. The second of these equations has solutions y = 0 or x = 1. Select the option that gives a complete list of the stationary points of the function. Select one: O (-3,0), (1, –4), (1, 4) O (0, -3), (-4, 1), (4, 1) O(1, -3), (-4, 0), (4,0) O(-3, 1), (0, -4), (0, 4)(3) The demand curve of a firm is p=1200– 21q and its total cost is C(q) = 2q° – 66q² +600q +1000 where q is the output of the firm (in thousands). Derive an expression R(q), for the firm' s revenue function. (i) (ii) Derive an expression II(g) for the firm' s profit function. (iii) Is the rate of change of profit increasing or decreasing when the output level of the firm is 10,000 units? (iv) Determine the level of output at which profit is maximized.1. x = t2+1, y = t3-1