31-35 Analytic Functions. Find f(2)= u(x, y) + iv(x, y) with u or u as given. Check for analyticity. 32. v e-3x sin 3y
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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.Q4: Find and classify all critical points for the function: z2 = (x3/3)-(y3/3)+2xyFind the linearization of the function f(x)=(1/3x+2) at x=−1. L(x)= x The exact value f(x) The appoximate value L(x) -1.1 -1.01 -1.001
- Finad the linearization at the given points. f(x)= 6/x^2 at (2, 3/2)3) Determine the domain and range of the following functions i) f (x,y) =x^2 + y^2 + 2 ii) f (x,y)= (y+1)/x iii) f (x,y) =√2-x+y iv) f (x,y) = cos (x-y)a. Let y=f(x) be the particular solution to the given differnetial equation with the initial condition f(2)=1. Write an equation for the line tangent to the graph of y=f(x) at x=2. Use your equation to approximate f(2.1). b. Find the particular solution to the given differnetial equation with the initial condition of f(2)=1.
- 3. The function f(x) = x^4 − 4x^3 + 8x has a critical point at x = 1.(a) Find f"(x). (b) Then use the second derivative test to identify the critical point as eithera local minimum, a local maximum, or neither.1. Find the fixed point(s) of f : [1 ;+infinity) follows [2 ;+infinity) defined by f(x) = square root of (x2 + x).Consider a differentiable function f with domain R and derivativesf'(x)=-aebx(1+bx) and f"(x)=-abebx(2+bx) , with a and b nonzero real numbers.The function has only one critical point x=-1/b and a local maximum at x=-1/bUse the Second Derivative test to find the value(s) of a and b
- For the function f(x) x3/3 - 3/4 x2 - 27/2 x + 39 1. determine the coordinates of the turning oints. 2. determine the coordinates for the point of inflection if it does exist. 3. what is the range of value overvwhich the functionis decreasing? 4. what is the range of values over which the function is strictly concave downwards?Let f(x,y)= 6-3xy+x2 a) The function f has only one stationary point. Find it. b) Is the stationary point of f a local maximum, or local minimum, or a saddle point? Justify your answer.Find a function of two variables f(x,y) for which D = 0, but the function has a minimum. Exclude the trivial functions f(x,y) = constant. Explain why your example has a minimum and show in detail that D = 0 at that minimum.