4) IVAN TUJU.7. LT If y = f(x) and y = g(x) are differentiable functions with...
Question 2 (20 points): Consider the functions f(x, y)-xe y sin y and g(x, y)-ys 1. Show f is differentiable in its domain 2. Compute the partial derivatives of g at (0,0) 3. Show that g is not differentiable at (0,0) 4. You are told that there is a function F : R2 → R with partial derivatives F(x,y) = x2 +4y and Fy(x, y 3x - y. Should you believe it? Explain why. (Hint: use Clairaut's theorem) Question 2...
Let H=F(x,y) and x=g(s,t), y=k(s,t) be differentiable functions. Now suppose that g(1,0)=8, k(1,0)=4, gs(1,0)=8, gt(1,0)=2, ks(1,0)=1, kt(1,0)=5, F(1,0)=9, F(8,4)=3, Fx(1,0)=13, Fy(1,0)=7, Fx(8,4)=9, Fy(8,4)=2. Find Hs(1,0), that is, the partial derivative of H with respect to s, evaluated at s=1 and t=0.
For problems 8-12, use the graph of y=f(x) and the table for g(x) and g'(x) to compute the indicated derivatives. Write your final answer and only your final answer) in the space provided. Answers should be exact and fractions should be used where appropriate (do not use numbers in decimal form). 1 -4 -2 g(x) 2 5/2 3 14/5 &'(x) 7/5 1/2 1/4 -1/4 0 2 قيا 2 - 1 -2 - 1/2 4 0 5 6 8 1 6...
5) Let P(1,2,2) be a point, and f(x,y,z) and g(x,y,z) be two differentiable functions satisfying the following conditions. 1) f(P)=1 and g(P)=4 og IT) = -2 Oz IP III) The direction in which f increases most rapidly at the point Pis ū=4i - +8k , and the derivative in this direction is 3. IV) Equation of the plane tangent to the surface f(x,y,z)+3g(x,y,z)=13 at the int P is x+4y + 5z =19 According to this, calculate og Ox . (20P)
3. Below you are given the graphs of the functions f and g. Suppose that: u(x)-f(g(x)), v(x)-f(x) g(x), and w(x)-g(f(x)) Use the graphs to find the indicated derivatives. If the indicated derivative does not exist, write "D.N.E." in the space provided. Be sure to include work that shows how you arrived at your answer. 20 a) u'3) b) v-4) c) wl) 3. Below you are given the graphs of the functions f and g. Suppose that: u(x)-f(g(x)), v(x)-f(x) g(x), and...
F. Te QB 2. (10 points) Let f and g be differentiable functions of x, and c an arbitrary positive constant number. Find the derivative of the function tanx h(x) - Vc + f(x) sec x Use the f' and g' notation. Size 40.72 KB g(x) Well
If z = f(x,y), where f is differentiable, and x = g(t) y = hết) g(3) = 2 h(3) = 7 g'(3) = 5 h'(3) = -4 fx(2,7) = 6 fy(2,7) = -8 Find dz/dt when t = 3.
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...
(8) Let E c R" and G C Rm be open. Suppose that f E -G and g:GR', so that h -gof:E R'. Prove that if f is differentiable at a point x E E and if g is differentiable at f(x) є G, then the partial derivatives Dh,(x) exist, for all , SO , . . . , n, and and J-: に1 The subscripts hi, 9i, k denote the coordinates of the functions h, g, f relative to...
please explain your solution with details. 7) Let f and g be differentiable functions such that 2< f(x)<4 and 2 s g(x)< 4 for all x. a) Find good upper and lower bounds on the arc O to x 4.(5 pts) length of the graph of f(x) from x= Ax,2 494, ANS L4 . Are Cang th Say tnt 1 + We can b) Can we find a good lower bound on the length of g(x) from x 0 to...