Option C is correct.... it the directional derivative with respect to e of F at x .
Just refer the limit formula of derivative ..
The partial derivative Let ei denote the ith standard basis vector of R The ith partial derivativ...
= Problem 2: Let S {ei, C2, C3} denote the standard basis of R3 and let B = {(1,0,0)*, (1,1,0), (1,1,1)t}. Find the matrices for the change of basis from S to B and its inverse. That is find Ibs and Isb
(8) 2 points Let f be a function defined and continuous, with continuous first partial derivative at the origin (0,0). A unit vector u for which D.f (0,0) is the maximum is: maximum a 1 (0,0)), A. /(0,0)x,0),y (0 af B. (0,0) 8x0,0),(0,0)), af 1 ((0,0),-y C. (0,0), /(0,0) D. None of the above.
(8) 2 points Let f be a function defined and continuous, with continuous first partial derivative at the origin (0,0). A unit vector u for which...
let P3 denote the vector space of polynomials of degree 3 or
less, with an inner product defined by
14. Let Ps denote the vector space of polynomials of degree 3 or less, with an inner product defined by (p, q) Ji p(x)q(x) dr. Find an orthogo- nal basis for Ps that contains the vector 1+r. Find the norm (length) of each of your basis elements
14. Let Ps denote the vector space of polynomials of degree 3 or less,...
(18) Let f and g be functions from R to R that have derivatives of al orders. Let h(k) denote the kth derivative of any function. Prove using the product rule for derivatives, the fact that and induction that k +1 k=0 (19) The Fibonacci numbers are defined recursively by Fn+2 = Fn+1 Prove that the number of subsets of { 1, 2, 3, . . . , n} containing no two successive integers is E, (20) Prove that 7n...
Consider a subset alpha={x+x2,1+x2,1 2x+2x2}ofP2(R). (a) Show
that alpha is a basis for P2(R). (b) For f(x) = 1 + x + x2 2 P2(R),
find its coordinator vector [f] alpha with respect to alpha. (c)
Let = {1, x, x2} be the standard basis for P2(R), and let f(x) = a
+ bx + cx2 and g(x) = p+qx+rx2 be the elements of P2(R) and k 2 R.
Prove that [f+g] = [f] +[g] and [kf] = k[f] and...
Let V be the vector space consisting of all functions f: R + R satisfying f(x) = a exp(x) +b cos(x) + csin(x) for some real numbers a, b, and c. (The function exp refers to the exponential, exp(22) = e.) Let F be the basis (exp cos sin of V. Let T :V + V be the linear transformation T(f) = f + f' + 2f" (where f' is the derivative of f). You may use the linearity of...
Analysis problem
(b) Let f, q be defined on A to R and let c be a cluster point of A i. Show that if both lim f and lim (f + g) exist, then lim g exists. c I>c ii. If lim f and lim fg exist, does it follow that lim g exists? -c (c) Suppose that f and g have limits in R as x -> o and that f(x) < g(x) for all x € (a,...
Problem 1. Given the vector space P the basis B -<1,7,',r'> of P, let U - span[1,2]. V-span c and W -spanr x '] Which of the following statements is true? 1. UV = 0 2. UUV is a vector subspace of P -P 3. U nW - and for any vector subspace P of P UW SPP 4. UUW = P. 5. All except statement 3 is false. Problem 2. Consider the function P, R such that f(1-r) -...
vector calculus.
Do
both please
Q1: What, in your own words, is the difference between a partial derivative and a directional derivative? How are they similar? Give a particular example to illustrate your explanations (choose some function z=f(x,y)) Q2: Given a surface z= f(x,y), a point (x,yo) in the domain of f, and a unit vector i pointing some direction in the xy-plane, what does it mean if D,f(x,Y)=0? Be as specific as possible.
Q1: What, in your own words,...
In the vector space R, let 8 {(1,3,0), (1, -3, 0), (0, 2, 2)}. (a) (6 points) Show that y is a basis of R3. (b) (7 points) Find the matrix [I,where I is the identity transform R3 R3 (c) (7 points) Using the matrix [I, convert the vector (r, y, z) into coordinates with respect to y instead of B. In other words, find ((x, y, z)] {(1,0,0), (0, 1,0), (0,0, 1)} be the standard basis, and let