Exercise 1. Do the following: (a) Write a statement defining the Chain Rule for the functions...
Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show that gr) F(x) :=| f(t)dt Jg(a) is differentiable and compute its derivative Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show that gr) F(x) :=| f(t)dt Jg(a) is differentiable and compute its derivative
please show all work, even trivial steps. Here are definitions if needed. do not write in script thank you! 4. Letf: R2 → R2, by f(x,y) = (x-ey,xy). a. Find Df (2,0). b. Find DF-1(f (2,0)) Inverse Function Theorem: Suppose that f:R" → R" is continuously differentiable in an open set containing a and det(Df(a)) = 0, then there is an open set, V, containing a and an open set, W, containing f(a) such that f:V W has a continuous...
a. Use the Chain Rule to find the indicated partial derivatives. z = x4 + x2y, x = s + 2t − u, y = stu2; ∂z ∂s ∂z ∂t ∂z ∂u when s = 1, t = 2, u = 3 b. Use the Chain Rule to find the indicated partial derivatives. w = xy + yz + zx, x = r cos(θ), y = r sin(θ), z = rθ; ∂w ∂r ∂w ∂θ when r = 8, θ = pi/2 c. Use the...
1. Determine whether the statement is true or false. If false, explain why and correct the statement (T/FIf)exists, then lim ()f) o( T / F ) If f is continuous, then lim f(x) = f(r) (TFo)-L, then lim f(x)- lim F(x) "( T / F ) If lim -f(x)s lim. f(x) L, then lim f(x)s 1. "(T/F) lim. In x -oo . (T/F) lim0 ·(T / F ) The derivative f' (a) is the instantaneous rate of change of y...
Whats the answer to number 1? 1. Let r(t) = -i-e2t j + (t? + 2t)k be the position of a particle moving in space. a. Find the particle's velocity, speed and direction at t = 0. Write the velocity as a product of speed and direction at this time. b. Find the parametric equation of the line tangent to the path of the particle at t = 0. 2. Find the integrals: a. S (tezi - 3sin(2t)j + ick)...
what is the answer for number 4 1. Let r(t) = -i-e2t j + (t? + 2t)k be the position of a particle moving in space. a. Find the particle's velocity, speed and direction at t = 0. Write the velocity as a product of speed and direction at this time. b. Find the parametric equation of the line tangent to the path of the particle at t = 0. 2. Find the integrals: a. S (tezi - 3sin(2t)j +...
Exercise 1. Tangent plane (15 pts) Let (S) be the surface given by the following equation. x+y2 = 1 + z2 An equation of the tangent plane to (S) at A(1,2,2) is: a. 2x + 4y - 4z = 1 b. x +y - z = 0 c. x + 2y – 2z = 1 d. x + y -z = 2 e. None of the above a b d. Exercise 3. Chain rule (15 pts) Let f(x,y,z) = xy...
Let Coo denote the set of smooth functions, ie, functions f : R → R whose nth derivative exists, for all n. Recall that this is a vector space, where "vectors" of Coo are function:s like f(t) = sin(t) or f(t) = te, or polynomials like f(t)-t2-2, or constant functions like f(t) = 5, and more The set of smooth functions f (t) which satisfy the differential equation f"(t) +2f (t) -0 for all t, is the same as the...
Problem 4.9 (e) /(z) = and γ is parametrized by r(t), 0 z + t 1, and satisfies Imr(t)> 0, r(0) -4 + i, and γ(1) 6 + 2i (f) f(s) sin(z) and γ is some piecewise smooth path from 1 to π. 4.2 and the fact that the length of γ does not change under 4.9. Prove Proposi reparametrization. (Hint: Assume γ, σ, and τ are smooth. Start with the definition off, f, apply the chain rule to σ...
#2 please Econ303 Homework 1B oints Find the derivatives of the following functions. (a) f(x) a + b (b) f(a) V5 +7 (c) f(x) (d) f(x) (x +3)(3-9x +11) 2. 3 p oints The following function describes how the value of import of country country j, Fiy is determined: In Fy 14.44+0.8521nG, +0.178 In G, -1.119 In Diy where Gi is GDP in country i, Gj GDP in country j, and D the two countries. ij the distance between Economists...