#7, (a) Show lim (x+1)y does not exist. (x+1)²=42 that (x,y) + (-1,0) x+y²+1 (b) Given p(x, y, z) = ex²-4 + z ln (1+ y²) -1/2 where yzz x=u²=uv+~~ | y=W+v² 5, z=w²+V-u use the chain rule to find op (in terms ού x, y, z, u, v, w (C) Evaluate of when (u,v, w) = (1,2,0).

Let a function g be defined on R2 by g(x,y) = (x2yesin(xy^2))/(x2+y2) use g(x,0), g(x,kx), g(x,kx2) , g(ly2,y), where k and l are constants (k =/ 0) and (l =/ 0), and g(rcosθ,rsinθ) to determine if lim(x,y)-->(0,0) g(x,y) exists  Also use the squeeze technique to the determine if the limit exists

e) f'(x)=(f.fx) 8. Let f(x)= a) (fog)(x) c) (fof)(x) - a) y = x² +1 9. Find the domain of each function. and g(x) = 3x+1, find the following: b) (gof)(x) b) y Q Search 2 x-3 f) (fog)(x) O d) (gog)(x) 10. What is the slope of the line passing through (-5, 6) and (1, -3)? Then find the equation of this line. Write the equation in slope-intercept form. (2pts) c) y = √x +4 me 31 10 MIL UT

4. Let H be a Hilbert space, YcHa subspace of H.

The period of the function f(x) = 3sin(x-) is %3D (v) 2n (a) 2 (c) 27 (d): 3 إختر أحد الخيارات: (c) O (a) O (d) O (b) O

3x Use the Quotient Rule to differentiate the function f(x)= x' +5 3(-5+4x°) a. f'(x)= – 3(-5-4x*) (1$ + s)° b. f'(x)= 3(5+ 5x) c. f'(x)= (15 + 5)² 3(5 + 4x°) (r$ + 5)? d. f'(x) = %3D 3(5+ 6x°) (r° + s)² e. f'(x) = -

c) Consider the third-order truncation error (O(Ax )³) with the step size Ax and also (-Ax ) then, subtract them and finally obtain a finite difference approximation for the first-order derivative of U respect x (Ux(x, t)). d) Consider the fourth-order truncation error (0(Ax )*) with the step size Ax and also (-Ax) then, add them and finally obtain a finite difference approximation for the second-order derivative of U respect x (Uxx(x, t)).

Vx - Vy+9 lim = (х, у)--(11,2) х+у+9 х — у — 9

Let f(x)= 1/(3x+2) find the values below 1) f(x+h)= 2) (f(x+h)-f(x))= 3) lim h→0 (f(x+h)−f(x))/h 4) Find the equation of the line tangent to the graph of f at x=2.  y=?

Consider a fumction f whose first and second derivatives are f(r) = 3. and f"(x)= (Ax² +1)*/2 (4x²+1)%/2 Which one of the following statement is the most accurate. (A) The function f is concave upward everywhere. (B) The function f is concave downward everywhere. (C) There is non inflection for the graph of f. (D) There is two inflection points at r= 0 and at r= 1. (E) The function f is concave upward on (-x, 0) and concave downward on (0, o0). (F) The function f is concave downward on (-o, 0) and concave upward on (0, 0).