7. [5 marks] Suppose that f(x), g(x), and h(x) are functions such that f(x) is O(g(x))...
Differentiate. Let f and g be functions that satisfy: f(4)-1, g(4)--3, f'(4)--2,and g'(4)-3. Finod h(4) for h(x)-f(x)g(x)-2/(x)+'7 O-5 -13 13 Differentiate. Let f and g be functions that satisfy: f(4)-1, g(4)--3, f'(4)--2,and g'(4)-3. Finod h(4) for h(x)-f(x)g(x)-2/(x)+'7 O-5 -13 13
Suppose f and g are continuous functions such that g(7) = 5 and such that f is defined at x = 7. Assume also that lim 5 f(x) + f(x)g(x)] = 20. Find f(7). f(7) =
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)...
Express the given function h as a composition of two functions f and g so that h(x)=(f o g)(x) h(x) = 1/2x-2 Choose the correct pair of functions. Find the donain of the function. What is the domain of g? (Type your answer in interval notation.)
4. (3 marks) Suppose that you have two linear functions f(x) and g(z) such that f(a) = g(a) = 0. Suppose also that f(k) = 6 and g(k) 3 for some ke R (see figure below). Determine f(x) 6 3
S f(r)da= g(x)dz. Prove a,bsuch that (8) Suppose f and g are continuous functions on that there is ro e (a, b) such that f(ro) = g(xo). (9) Prove that if the function f is continuous on a, b, then there is c E [a, b such that f(x)dax a Ja f(e) S f(r)da= g(x)dz. Prove a,bsuch that (8) Suppose f and g are continuous functions on that there is ro e (a, b) such that f(ro) = g(xo). (9)...
Suppose that G,H are groups with identities eG,eH, respectively. Define f: G x H→G by setting, for all g in G and h in H, f(g,h) = g. a) Prove that f is a homomorphism. b) Prove that f is surjective. c) Prove that ker(f) = {(eG,h) | h in H}.
Let {h} be a sequence ofRiemann integrable functions on [a,b], such that for each x, {h(x)) is a decreasing sequence. Suppose n) converges pointwise to a Riemann integrable function f Prove that f(x)dxf(x)dx. lim n00 Let {h} be a sequence ofRiemann integrable functions on [a,b], such that for each x, {h(x)) is a decreasing sequence. Suppose n) converges pointwise to a Riemann integrable function f Prove that f(x)dxf(x)dx. lim n00
(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....
(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...