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3. Let fRR' and g:R R2 be given by a) Write down the derivative matrices g'(u)...
3. Let f RR2 and g RR2 be given by 2x1323 u1u2 T3 a) Write down the derivative matrices g' (u) an...
show all steps, please.
3. Let f RR2 and g RR2 be given by 2x1323 u1u2 T3 a) Write down the derivative matrices g' (u) and f'(a) and use the chain rule to find the derivative matrix (g o f)'(x); b) Are the entries of the new function (go f)(x) a linear or nonlinear function of ? 1 3 marks mark c) How do you understand the statement "(Df) (x) is a linear function" in Section 4.1 of the Class...
question starts at let. than one variable. Let f:R? → R3 be the function given by...
question starts at let.
than one variable. Let f:R? → R3 be the function given by f(x, y) = (cos(x3 - y2), sin(y2 – x), e3x2-x-2y). (a) Let P be a point in the domain of f. As we saw in class, for (x, y) near P, we have f(x, y) f(P) + (Dpf)(h), where h = (x, y) - P. The expression on the right hand side is called the linear approximation of f around P. Compute the linear...
4. Consider the functions f : R2 R2 and g R2 R2 given by f(x, y) (x, xy) and g(x, y)-(x2 + y, x +...
4. Consider the functions f : R2 R2 and g R2 R2 given by f(x, y) (x, xy) and g(x, y)-(x2 + y, x + y) (a) Prove that f and g are differentiable everywhere. You may use the theorem you stated in (b) Call F-fog. Properly use the Chain Rule to prove that F is differentiable at the point question (1c). (1,1), and write F'(1, 1) as a Jacobian matrix.
4. Consider the functions f : R2 R2 and...
(1 point) 5x2 — 5у, v %3D 4х + Зу, f(u, U) sin u cos v,u...
(1 point) 5x2 — 5у, v %3D 4х + Зу, f(u, U) sin u cos v,u = Let z = = and put g(x, y) = (u(x, y), v(x, y). The derivative matrix D(f ° g)(x, y) (Leaving your answer in terms of u, v, x, y ) (1 point) Evaluate d r(g(t)) using the Chain Rule: r() %3D (ё. e*, -9), g(0) 3t 6 = rg() = dt g(u, v, w) and u(r, s), v(r, s), w(r, s). How...
Question 5, Let ơ (u, v) : R2- R3 be a smooth function (not necessarily a surface patch). Let E O...
please be as detailed as possible
Question 5, Let ơ (u, v) : R2- R3 be a smooth function (not necessarily a surface patch). Let E Ou .Ou, F-Ou . συ and G Oy .Oy. Show that the following equalities hold: (Here D denotes total derivative.)
Question 5, Let ơ (u, v) : R2- R3 be a smooth function (not necessarily a surface patch). Let E Ou .Ou, F-Ou . συ and G Oy .Oy. Show that the following equalities...
Find the derivative of the function. y=5 e 2x2-3 dy 80 e 2x2x2 * + 20...
Find the derivative of the function. y=5 e 2x2-3 dy 80 e 2x2x2 * + 20 e 2x2 X X Try again. Differentiate the exponential using the Chain Rule. to le 9(x) = g(x)g'(x) OK Find the derivative of the given function. y = (3x2 + 5x) -7x e -7x -7x y' = -21 e -7xx2 -29 +5e (Type an exact answer.) x X That's incorrect. The product rule states that if f(x) = u(x)•v(x), and if both u'(x) and...
5. Let f R2 ->R2 be the function given by f(x, y) (х + у, х...
5. Let f R2 ->R2 be the function given by f(x, y) (х + у, х — у). (i) Prove that f is linear as a function from R2 to R2. (ii) Calculatee the matrix of f. (iii) Prove that f is a one-to-one function whose range is R2. Deduce that f has an inverse function and calculate it. (iv) If C is the square in R2 given by C = [0,1] x [0, 1], find the set f(C), illustrating...
eca Problem 3. Write down f(x) = as a composition f = goh of an expo-...
eca Problem 3. Write down f(x) = as a composition f = goh of an expo- nential function g and a power function h. Then apply chain rule to compute derivative of f. (3+4 points) Bonus Points: For the functions g, h found in Problem 3 above, write down the composition h o g, composition of them in the other order. Call this new function F. Argue why F is not the same function as f. Then apply chain rule...
3. Show that (a) the function g: R” → R, given by g(x) = ||2||2, is...
3. Show that (a) the function g: R” → R, given by g(x) = ||2||2, is convex. (b) if f : RM → R is convex, then g:R" + R given by g(x) = f(Ax – b) is also convex. A here is an m x n matrix, and b ERM is a vector. You may use any of the results we covered in class (but the definition of convexity may be an easy way to do this, and gives...
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (...
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (i Determine the level curves for the surface when z on the same diagram in the r-y plane. 1 and 2, Sketch the level curves (i) Determine the cross-sectional curves of the surface in the r-z plane and in the y- plane. Sketch the two cross-sectional curves (iii) Sketch the surface. (b) For the point (r, y)...
show all steps, please.
3. Let f RR2 and g RR2 be given by 2x1323 u1u2 T3 a) Write down the derivative matrices g' (u) and f'(a) and use the chain rule to find the derivative matrix (g o f)'(x); b) Are the entries of the new function (go f)(x) a linear or nonlinear function of ? 1 3 marks mark c) How do you understand the statement "(Df) (x) is a linear function" in Section 4.1 of the Class...
question starts at let.
than one variable. Let f:R? → R3 be the function given by f(x, y) = (cos(x3 - y2), sin(y2 – x), e3x2-x-2y). (a) Let P be a point in the domain of f. As we saw in class, for (x, y) near P, we have f(x, y) f(P) + (Dpf)(h), where h = (x, y) - P. The expression on the right hand side is called the linear approximation of f around P. Compute the linear...
4. Consider the functions f : R2 R2 and g R2 R2 given by f(x, y) (x, xy) and g(x, y)-(x2 + y, x + y) (a) Prove that f and g are differentiable everywhere. You may use the theorem you stated in (b) Call F-fog. Properly use the Chain Rule to prove that F is differentiable at the point question (1c). (1,1), and write F'(1, 1) as a Jacobian matrix.
4. Consider the functions f : R2 R2 and...
(1 point) 5x2 — 5у, v %3D 4х + Зу, f(u, U) sin u cos v,u = Let z = = and put g(x, y) = (u(x, y), v(x, y). The derivative matrix D(f ° g)(x, y) (Leaving your answer in terms of u, v, x, y ) (1 point) Evaluate d r(g(t)) using the Chain Rule: r() %3D (ё. e*, -9), g(0) 3t 6 = rg() = dt g(u, v, w) and u(r, s), v(r, s), w(r, s). How...
please be as detailed as possible
Question 5, Let ơ (u, v) : R2- R3 be a smooth function (not necessarily a surface patch). Let E Ou .Ou, F-Ou . συ and G Oy .Oy. Show that the following equalities hold: (Here D denotes total derivative.)
Question 5, Let ơ (u, v) : R2- R3 be a smooth function (not necessarily a surface patch). Let E Ou .Ou, F-Ou . συ and G Oy .Oy. Show that the following equalities...
Find the derivative of the function. y=5 e 2x2-3 dy 80 e 2x2x2 * + 20 e 2x2 X X Try again. Differentiate the exponential using the Chain Rule. to le 9(x) = g(x)g'(x) OK Find the derivative of the given function. y = (3x2 + 5x) -7x e -7x -7x y' = -21 e -7xx2 -29 +5e (Type an exact answer.) x X That's incorrect. The product rule states that if f(x) = u(x)•v(x), and if both u'(x) and...
5. Let f R2 ->R2 be the function given by f(x, y) (х + у, х — у). (i) Prove that f is linear as a function from R2 to R2. (ii) Calculatee the matrix of f. (iii) Prove that f is a one-to-one function whose range is R2. Deduce that f has an inverse function and calculate it. (iv) If C is the square in R2 given by C = [0,1] x [0, 1], find the set f(C), illustrating...
eca Problem 3. Write down f(x) = as a composition f = goh of an expo- nential function g and a power function h. Then apply chain rule to compute derivative of f. (3+4 points) Bonus Points: For the functions g, h found in Problem 3 above, write down the composition h o g, composition of them in the other order. Call this new function F. Argue why F is not the same function as f. Then apply chain rule...
3. Show that (a) the function g: R” → R, given by g(x) = ||2||2, is convex. (b) if f : RM → R is convex, then g:R" + R given by g(x) = f(Ax – b) is also convex. A here is an m x n matrix, and b ERM is a vector. You may use any of the results we covered in class (but the definition of convexity may be an easy way to do this, and gives...
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (i Determine the level curves for the surface when z on the same diagram in the r-y plane. 1 and 2, Sketch the level curves (i) Determine the cross-sectional curves of the surface in the r-z plane and in the y- plane. Sketch the two cross-sectional curves (iii) Sketch the surface. (b) For the point (r, y)...
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