Question

Let z=f(a,b,c) where a=g(s,t), b=h(l(s+t),t), c=tsin(s). f,g,h,l are all differentiable functions. Compute the partial derivatives of z with respect to s and the partial of z with respect to t.

Answer #1

Suppose that f is a twice differentiable function and that
its second partial derivatives are continuous. Let h(t) =
f (x(t), y(t)) where x = 2e^ t and y = 2t. Suppose that
fx(2, 0) = 1, fy(2, 0) = 3, fxx(2, 0) = 4, fyy(2, 0) = 1, and
fxy(2, 0) = 4. Find d ^2h/ dt ^2 when t = 0.

Suppose that f is a twice differentiable function and that
its second partial derivatives are continuous. Let h(t) =
f (x(t), y(t)) where x = 3e ^t and y = 2t. Suppose that
fx(3, 0) = 2, fy(3, 0) = 1, fxx(3, 0) = 3, fyy(3, 0) = 2, and
fxy(3, 0) = 1. Find d 2h dt 2 when t = 0.

Assume that all the given functions have continuous second-order
partial derivatives. If z = f(x, y), where x = r2 + s2 and y = 6rs,
find ∂2z/∂r∂s. (Compare with Example 7.) ∂2z/∂r∂s = ∂2z/∂x2 +
∂2z/∂y2 + ∂2z/∂x∂y + ∂z/∂y

2. (a)
Determine all first and second order partial derivatives of
f(x,y,z) = x2y3 sin(xz)
(b) Determine all first-order partial derivatives of
g(x,y,z)=u2y+x2v where u=exz,
v=sin(yz)

Let S = {a,b,c,d,e,f,g} and let T = {1,2,3,4,5,6,7,8}.
a. How many diﬀerent functions are there from S to
T?
b. How many diﬀerent one-to-one functions are there from S to
T?
c. How many diﬀerent one-to-one functions are there from T to
S?
d. How many diﬀerent onto functions are there from T to
S?

Let f(x,y,z)=xy+z^3, x=r+s−8t, y=3rt, z=s^6.
Use the Chain Rule to calculate the partial derivatives.
(Use symbolic notation and fractions where needed. Express the
answer in terms of independent variables

Let f(z) and g(z) be entire functions, with |f(z) - g(z)| < M
for some positive real number M and all z in C. Prove that f'(z) =
g'(z) for all z in C.

Let S be the set of all functions from Z to Z, and consider the
relation on S:
R = {(f,g) : f(0) + g(0) = 0}.
Determine whether R is (a) reﬂexive; (b) symmetric; (c)
transitive; (d) an equivalence relation.

1. Let A = {1,2,3,4} and let F be the set of all functions f
from A to A. Prove or disprove each of the following
statements.
(a)For all functions f, g, h∈F, if f◦g=f◦h then g=h.
(b)For all functions f, g, h∈F, iff◦g=f◦h and f is one-to-one
then g=h.
(c) For all functions f, g, h ∈ F , if g ◦ f = h ◦ f then g =
h.
(d) For all functions f, g, h ∈...

Let f and g be continuous functions from C to C and let D be a
dense
subset of C, i.e., the closure of D equals to C. Prove that if
f(z) = g(z) for
all x element of D, then f = g on C.

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