Graph of f
Graph of degree 2 approximated polynomial
1. Let f(x, y) 11x2T (a) Find the critical point of f (b) Compute the degree-two...
4. Let f()VI+ x. (a) Compute P2(x), the degree 2 Taylor polynomial for f at ro 0. (b) Use P2 to approximate f(0.5) required to evaluate a real polynomial of degree 5. How many multiplications number? Explain n at a real are 6. Show that if x, y and ry are real mumbers in the range of our floating point system, then ay-f(ry3 + O(*) ay
Let ?(?)=?2−8?+4f(x)=x2−8x+4. (1 point) Let f(x) = x2 – 8x + 4. Find the critical point c of f(x) and compute f(c). The critical point c is = The value of f(c) = Compute the value of f(x) at the endpoints of the interval [0, 8]. f(0) = f(8) = Determine the min and max Minimum value = Maximum value = Find the extreme values of f(x) on [0, 1]. Minimum value = Maximum value =
Given a two-variable function f(x, y), if P(x0,yo) is a critical point, then the behavior of f around P can be approximated by its second order terms according to Taylor series, that is, f(x,y) = f(P) + F(x – xo)?H (x, y) , where H(x, y) = fyy(P)(=%)2 + 2 fxy(P) (?=%) + fxx(P). (a). If H(x, y) > 0 for all x,y, is P a local max, local min or saddle point? (b). Let s = (4=90). Then, H(x,...
2. Let 6 marks (a) Find f(x),f"(x), and f"(x). (b) Find the second order Taylor expansion of f at 1, namely f(r) = ao + ala-1 ) + a2(z-1)2 + R2(x), where Ra is the remainder. You should find ao, a, a2, and R(p). 8 marks that the error in this estimation (i.e., R2(0.9)1) is at most 10-3. 6 marks (c) Use the Taylor expansion found above to estimate the value of f(0.9). Show Find f(x), f"(), and f" (b)...
5. Find the 2nd-order Taylor approximation of f(x, y) = el+22 –y? around the critical point (which you have to find) of f(x,y). Us- ing this approximation explain why the critical point is a saddle point. Hint: f(xo + h) = f(x0) + Df(xo)h + ihBh, where B is the matrix with elements on your ; i, j = 1, 2, ..., n.
Let f(x, y) = x(x – 1) + y2. (a) [1 point] Sketch the level curves of f. (b) [2 points] Compute the gradient of f, and sketch it as a vector field. (c) [3 points) Find all critical values of f and classify them as local maxima, local minima, or saddle points.
-n ', S Let f(x,yZFz2_xy. Let v=<1,1,1>. Let point P=<2,1,3> a. Compute gradient of fx,y,z) b. If the contours are far apart, is the length of the gradient large or small? Answer: Explain! What MATLAB command is used to draw the gradient vectors? Answer: - c. Compute the directional derivative in the direction of v. d. Compute the equation of the tangent plane to f(x,y,z) at the point P. e. Use the chain rule to compute r if x t2,...
5. Let f(x,y) = 3x2 y - y3 - 6x. (a) Find all the critical points off. (b) Classify each of the critical points; that is, what type are they? (c) For the same function f(x,y), find the maximum value of f on the unit square, 0 SX S1,0 <y s 1.
(1 point) Consider the function f(x) = xin(x). Let T, be the degree Taylor approximation of f(2) about x = 1. Find: T = T = Use 3 decimal places in your answer, but make sure you carry all decimals when performing calculations T3 is an (over/under) estimate of f(2). If R3 is the remainder given by the Lagrange Remainder Formula: |R3|
(1 point) Consider the function f(x) = Vx + 1. Let Tn be the nth degree Taylor approximation of f(10) about x = 8. Find: T = T2 = Use 3 decimal places in your answer, but make sure you carry all decimals when performing calculations T, is an (over/under) estimate of f(10). If R2 is the remainder given by the Lagrange Remainder Formula: |R2| =