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.
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,...
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) IVAN TUJU.7. LT If y = f(x) and y = g(x) are differentiable functions with values of the functions and their derivatives as indicated in the table below, compute the derivative (5(8(x)) + g(f(x))) evaluated at x = 4: * f(x) f(x) g(x) g(x) 4 5 8 8 11 5 4 12 8 1124 10
(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...
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...
Suppose fis a differentiable function of x and y, and g(r,s) - 8r - S, s2 - 7/). Use the table of values below to calculate 9:3, 6) and 9:(3,6). f fx 9 4 Fy 3 9 2 (18, 15) (3,6) 4 9 7 8 9.(3, 6) = 9s(3, 6) = Need Help? Read It Talk to Tutor
(8) Let E C R" and G C R" be open. Suppose that f E G and g G R', so that h = go f : E → R. Prove that if f is differentiable at a point x E E, and if g is differentiable at f (x) E G, then the partial derivatives Dihj(x) exist, for all and j - ...., and 7m に! (The subscripts hi. g. fk denote the coordinates of the functions h, g....
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
Assuming that the equation defines x and y implicitly as differentiable functions x =f(t), y =g(t), find the slope of the curve x =f(t), y=g(t) at the given value of t. x3 +41? = 37, 2y3 - 22 = 110, t = 3 The slope of the curve at t= 3 is (Type an integer or simplified fraction.)
7. Suppose We have three functions f(x), g(x), and h(x), such that f(-2) = 7, 9(-2) = 3, h(-2) = 10, f'(-2) = -14, 5'(-2) = 0, and '(-2) = 100. What is the derivative of In [Chooker)] at x = -2? a)-16 b) -0.22 c) -16.5 d) -33.5 e) -3/4 8. What is the slope of the tangent line (dy/dx) at the point (1,0) to the curve given by the equation (78 + y) = (1 - 4y)? a)...
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)