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Let γ(t) be a differentiable curve in R". If there is some differentiable function F :...
full workings required Let f: R^2 → be a differentiable function and let CCR be a...
full workings required
Let f: R^2 → be a differentiable function and let CCR be a curve in R^2 described by the cartesian equation f(x,y) = Letla.b) R be a point that lies on the curve Cck and assume that the partial derivatives off evaluated at (a,b) satisfy: fr(a,b) 0 and fy(a,b) +0. Also assume that there exists an expression y-g(x) that solves the equation f(xxx)=0 fory in terms of x in a neighbourhood of the point (8.b). This means...
Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r)) =...
Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r)) = K for some constant K is orthogonal to the tangent vector T() of each curve C described by the vector function on the surface passing through Po (xo,yo, zo). Hint, remember that the tangent vector T(o) R'(), so prove that Vfo R'O) 0
Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r))...
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
Example A.3 Surface normal vector. Let S be a surface that is represented by f(x, y, z) -c, where...
Example A.3 Surface normal vector. Let S be a surface that is represented by f(x, y, z) -c, where f is defined and differentiable in a space. Then, let C be a curve on S through a point P-Go, yo,Zo) on S, where C is represented by rt)[x(t), y(t), z(t)] with r(to) -[xo. Vo, zol. Since C lies on S, r(t) must satisfy f(x, y. z)-c, or f(x(t), y(t), z(t))-c. Show that vf is orthogonal to any tangent vector r'(t)...
Let f : Rn × Rn → R be the inner product function: f(r,y)-(2,3) 1. Using...
Let f : Rn × Rn → R be the inner product function: f(r,y)-(2,3) 1. Using the definition of multivariable derivative, calculate D fab and the Jacobian matrix f'(a, b) 2. If f, g : Rn → R are differentiable and h : R → R is defined by h(t)-(f(t), g(t)), show that 3. If f : R → Rn is differentiable and Ilf(t)ll = 1 for all t, show that(f,(t)T,f(t))-0
Let U be an open subset of R. Let f: U C Rn → Rm. (a) Prove that f is continuously differentiable if and only if for ea...
Let U be an open subset of R. Let f: U C Rn → Rm. (a) Prove that f is continuously differentiable if and only if for each a є U, for each E > 0, there exists δ > 0 such that for each x E U, if IIx-all < δ, then llDf(x)-Df(a) ll < ε. (b) Let m n. Prove that if f is continuously differentiable, a E U, and Df (a) is invertible, then there exists δ...
5. [4 bonus points) Let F be a continuously differentiable non-decreasing function on R with F'...
5. [4 bonus points) Let F be a continuously differentiable non-decreasing function on R with F' = f. Show that 1,6)dF(a) = 5,5(x)}(a)dx 2 A A for every non-negative function g on R and a Borel set A.
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)).
5, (25 points 4 pages max) Suppose that γ(t) = (x(t), y(t)) is a smooth (infinitely differentiabl...
5, (25 points 4 pages max) Suppose that γ(t) = (x(t), y(t)) is a smooth (infinitely differentiable) plane curve. For curves such that lh'(t) 0, the (signed) curvature is defined to be the quantity K(t) (a) Suppose the curve γ(t) is the graph of a function, ie x(t)-t and y(t) f(t) for some function f. Write the formula for the curve in this case. Suppose you were at a critical point of the graph of f. What does the curvature...
Consider the function Let where f(t) is differentiable for all t ∈ R. Show that z satisfies the ...
Consider the function Let where f(t) is differentiable for all t ∈ R. Show that z satisfies the partial differential equation (x2 − y2 ) ∂z/∂x + xy ∂z/∂y = xyz for all (x, y) ∈ R2 \ { (t, 0)|t ∈ R }.
full workings required
Let f: R^2 → be a differentiable function and let CCR be a curve in R^2 described by the cartesian equation f(x,y) = Letla.b) R be a point that lies on the curve Cck and assume that the partial derivatives off evaluated at (a,b) satisfy: fr(a,b) 0 and fy(a,b) +0. Also assume that there exists an expression y-g(x) that solves the equation f(xxx)=0 fory in terms of x in a neighbourhood of the point (8.b). This means...
Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r)) = K for some constant K is orthogonal to the tangent vector T() of each curve C described by the vector function on the surface passing through Po (xo,yo, zo). Hint, remember that the tangent vector T(o) R'(), so prove that Vfo R'O) 0
Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r))...
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
Example A.3 Surface normal vector. Let S be a surface that is represented by f(x, y, z) -c, where f is defined and differentiable in a space. Then, let C be a curve on S through a point P-Go, yo,Zo) on S, where C is represented by rt)[x(t), y(t), z(t)] with r(to) -[xo. Vo, zol. Since C lies on S, r(t) must satisfy f(x, y. z)-c, or f(x(t), y(t), z(t))-c. Show that vf is orthogonal to any tangent vector r'(t)...
Let f : Rn × Rn → R be the inner product function: f(r,y)-(2,3) 1. Using the definition of multivariable derivative, calculate D fab and the Jacobian matrix f'(a, b) 2. If f, g : Rn → R are differentiable and h : R → R is defined by h(t)-(f(t), g(t)), show that 3. If f : R → Rn is differentiable and Ilf(t)ll = 1 for all t, show that(f,(t)T,f(t))-0
Let U be an open subset of R. Let f: U C Rn → Rm. (a) Prove that f is continuously differentiable if and only if for each a є U, for each E > 0, there exists δ > 0 such that for each x E U, if IIx-all < δ, then llDf(x)-Df(a) ll < ε. (b) Let m n. Prove that if f is continuously differentiable, a E U, and Df (a) is invertible, then there exists δ...
5. [4 bonus points) Let F be a continuously differentiable non-decreasing function on R with F' = f. Show that 1,6)dF(a) = 5,5(x)}(a)dx 2 A A for every non-negative function g on R and a Borel set A.
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, (25 points 4 pages max) Suppose that γ(t) = (x(t), y(t)) is a smooth (infinitely differentiable) plane curve. For curves such that lh'(t) 0, the (signed) curvature is defined to be the quantity K(t) (a) Suppose the curve γ(t) is the graph of a function, ie x(t)-t and y(t) f(t) for some function f. Write the formula for the curve in this case. Suppose you were at a critical point of the graph of f. What does the curvature...
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