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Let f: R^2 → be a differentiable function and let CCR be a...
Let γ(t) be a differentiable curve in R". If there is some differentiable function F :...
Let γ(t) be a differentiable curve in R". If there is some differentiable function F : Rn R with F(γ(t)) C constant, show that DF(γ(t))T is orthogonal to the tangent vector γ(t).
Let U be an open subset of R". Let f:UCR"-R be differentiable at a E U. In this exercise you will prove that if ▽f(a) 0, then at the point a, the function f increases fastest in the direction...
Let U be an open subset of R". Let f:UCR"-R be differentiable at a E U. In this exercise you will prove that if ▽f(a) 0, then at the point a, the function f increases fastest in the direction of V f(a), and the maximum rate of increase is Vf(a)l (a) Prove that for each unit vector u e R" (b) Prove that if ▽/(a)メ0, and u = ▽f(a)/IV/(a) 11, then
Let U be an open subset of R". Let...
4. (a) Assume a function h is differentiable at some point to. Is it true that...
4. (a) Assume a function h is differentiable at some point to. Is it true that h is continuous on some open-neighbourhood of xo? Provide either a proof or a counterexample. (b) Let f be twice differentiable on R and assume that f" is continuous. Show that for all x ER S(x) = S(0) + s°C)x + [ (x - 1))"(dt. (C) Deduce that for any twice continuously differentiable function f on R and any positive x > 0, x...
The tangent plane at a point Po(f(uo.VO) 9 (uo.vo) h(uo,VO)) on a parametrized surface r(u,v) =...
The tangent plane at a point Po(f(uo.VO) 9 (uo.vo) h(uo,VO)) on a parametrized surface r(u,v) = f(u,v) i + g(u,v) j+h(u, v) k is the plane through P, normal to the vector ru (uo.VO) XIV(40.VO) the cross product of the tangent vectors ru (uo. Vo) and rv (uo.VO) at Pg. Find an equation for the plane tangent to the surface at Po. Then find a Cartesian equation for the surface and sketch the surface and tangent plane together. (573 15...
Convex Optimization Let f: R R be a differentiable function on R. Show that f is...
Convex Optimization
Let f: R R be a differentiable function on R. Show that f is convex iff f' is nondecreasing (i.e. x y f'(x) <f'(y)).
let f: R —> R be differentiable & let f'(x) ≥ 4 for every real number...
let f: R —> R be differentiable & let f'(x) ≥ 4 for every real number x. if f(0) = 0 then show that f(2) ≥ 8
5, ( 10 pts.) Let f : R → R be a differentiable function and suppose that 2 for all xE R. Prove t...
5, ( 10 pts.) Let f : R → R be a differentiable function and suppose that 2 for all xE R. Prove that the equation f(cos) cos(f()) has a unique solution in R.
5, ( 10 pts.) Let f : R → R be a differentiable function and suppose that 2 for all xE R. Prove that the equation f(cos) cos(f()) has a unique solution in R.
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint:...
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem)
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
5. Let f : R -R be a differentiable function, and suppose that there is a constant A < 1 such tha...
5. Let f : R -R be a differentiable function, and suppose that there is a constant A < 1 such that If,(t)| < A for all real t. Let xo E R, and define a sequence fan] by 2Znt31(za),n=0,1,2 Prove that the sequence {xn) is convergent, and that its limit is the unique fixed point of f.
5. Let f : R -R be a differentiable function, and suppose that there is a constant A
Exercise 1. Let f : R R be differentiable on la, b, where a, b R...
Exercise 1. Let f : R R be differentiable on la, b, where a, b R and a < b, and let f be continuous on [a, b]. Show that for every e> 0 there exists a 6 > 0 such that the inequality f(x)- f(c) T-C holds for all c, x E [a, 히 satisfying 0 < |c-x| < δ
Let γ(t) be a differentiable curve in R". If there is some differentiable function F : Rn R with F(γ(t)) C constant, show that DF(γ(t))T is orthogonal to the tangent vector γ(t).
Let U be an open subset of R". Let f:UCR"-R be differentiable at a E U. In this exercise you will prove that if ▽f(a) 0, then at the point a, the function f increases fastest in the direction of V f(a), and the maximum rate of increase is Vf(a)l (a) Prove that for each unit vector u e R" (b) Prove that if ▽/(a)メ0, and u = ▽f(a)/IV/(a) 11, then
Let U be an open subset of R". Let...
4. (a) Assume a function h is differentiable at some point to. Is it true that h is continuous on some open-neighbourhood of xo? Provide either a proof or a counterexample. (b) Let f be twice differentiable on R and assume that f" is continuous. Show that for all x ER S(x) = S(0) + s°C)x + [ (x - 1))"(dt. (C) Deduce that for any twice continuously differentiable function f on R and any positive x > 0, x...
The tangent plane at a point Po(f(uo.VO) 9 (uo.vo) h(uo,VO)) on a parametrized surface r(u,v) = f(u,v) i + g(u,v) j+h(u, v) k is the plane through P, normal to the vector ru (uo.VO) XIV(40.VO) the cross product of the tangent vectors ru (uo. Vo) and rv (uo.VO) at Pg. Find an equation for the plane tangent to the surface at Po. Then find a Cartesian equation for the surface and sketch the surface and tangent plane together. (573 15...
Convex Optimization
Let f: R R be a differentiable function on R. Show that f is convex iff f' is nondecreasing (i.e. x y f'(x) <f'(y)).
5, ( 10 pts.) Let f : R → R be a differentiable function and suppose that 2 for all xE R. Prove that the equation f(cos) cos(f()) has a unique solution in R.
5, ( 10 pts.) Let f : R → R be a differentiable function and suppose that 2 for all xE R. Prove that the equation f(cos) cos(f()) has a unique solution in R.
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem)
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
5. Let f : R -R be a differentiable function, and suppose that there is a constant A < 1 such that If,(t)| < A for all real t. Let xo E R, and define a sequence fan] by 2Znt31(za),n=0,1,2 Prove that the sequence {xn) is convergent, and that its limit is the unique fixed point of f.
5. Let f : R -R be a differentiable function, and suppose that there is a constant A
Exercise 1. Let f : R R be differentiable on la, b, where a, b R and a < b, and let f be continuous on [a, b]. Show that for every e> 0 there exists a 6 > 0 such that the inequality f(x)- f(c) T-C holds for all c, x E [a, 히 satisfying 0 < |c-x| < δ
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