Given f(x, y) = sin(x + y) where x = s5t², y = 5s-2t. Find fs(r(s, t), y(s, t)) = f(x(s, t), y(s, t)) = Note: This question is looking for the answer to be only in terms of s and t.
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- Decay of Litter Litter such as leaves falls to the forest floor, where the action of insects and bacteria initiates the decay process. Let A be the amount of litter present, in grams per square meter, as a function of time t in years. If the litter falls at a constant rate of L grams per square meter per year, and if it decays at a constant proportional rate of k per year, then the limiting value of A is R=L/k. For this exercise and the next, we suppose that at time t=0, the forest floor is clear of litter. a. If D is the difference between the limiting value and A, so that D=RA, then D is an exponential function of time. Find the initial value of D in terms of R. b. The yearly decay factor for D is ek. Find a formula for D in term of R and k. Reminder:(ab)c=abc. c. Explain why A=RRekt.Chain Rule with two independent variables Let z = sin 2x cos 3y, where x = s + t and y = s - t. Evaluate ∂z/∂s and ∂z/∂t.Chain Rule with one independent variable Use Theorem 15.7 to find the following derivatives. dz/dt, where z = (x + 2y)10, x = sin2 t, y = (3t + 4)5
- y=A x ln(x) + B x + F is the particular solution of the second-order linear DEQ: xy'' =8 where y'=2 at the point (9,8). Determine A,B,F. y=A x ln(x) + B x + F is also called an explicit solution. Is the DEQ separable, exact, 1st-order linear, Bernouli?The characteristic equation for a 5th degree, homogeneous, constant coefficient DE is p(x)=x^3(x^2+1). Which linearly independent functions will be used to form the general solution? A. 1,x,x^2,cos(x),sin(x) B. 1,e^{-x},xe^{-x}, cos(x),sin(x) C. x,x^2,x^3,cos(x),sin(x) D. 1,e^x,xe^x, cos(x),sin(x)The number of defects on the front side (X) of a wooden panel and the number of defectsa on the rear side (Y) of the panel are under study. Suppose that the joint pmf of X and Y is modeled as fxy(x,y)=c(x+y), x= 1,2,3 and y=1,2,3. Determine the value of c.
- (a) Show that ∫-∞∞f(x) dx is divergent.(b) Show that Iim ∫-t t x dx = 0 t→∞ This shows that we can’t define ∫-∞∞f(x) dx = Iim ∫-t t f(x) dx = 0 t→∞find ƒx , ƒy , and ƒz . ƒ(x, y, z) = sec-1 (x + yz)Chain Rule with one independent variable Use Theorem 15.7 to find the following derivatives. dz/dt, where z = x sin y, x = t2, and y = 4t3
- Chain Rule with one independent variable Use Theorem 15.7 to find the following derivatives. dw/dt, where w = cos 2x sin 3y, x = t/2, and y = t4Find the value of 0x/0z at the point (1, -1, -3) if the equation xz + y ln x - x2 + 4 = 0 defines x as a function of the two independent variables y and z andthe partial derivative exists.Exercises 77 and 78 are about the triangle shown here.Characterize the 6 equations by putting a tick marks, numbers, or letters on the appropriate boxes for the descriptions/requirements indicated in the table below.(please see the equation and table at the pictures) a) Obtain the particular solution of xydx + ( x + 5 y ) dy = 0; when x = 0, y = 1b) Find the general solution of (3sin y − 5x)dx + 2x2 cot ydy = 0 ( (x−y) )c) Obtain the particular solution of dy = 1+ 6xe dx; when x = 0, y = 0 d)Obtaintheparticularsolution of (2x−3y+4)dx+3(x−1)dy=0; whenx=−1, y=2 e) Find the general solution of y2dx + (3xy + y2 −1)dy = 0