Let I ⊆R be an open interval,let c ∈I,let f,g:I→R be functions and let k ∈R.Suppose that f and g are differentiable at c.
f-g is differntiable at c and |f-g|' (c)=f'(c)-g'(c)

Let I ⊆R be an open interval,let c ∈I,let f,g:I→R be functions and let k ∈R.Suppose...
(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...
(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....
2. Let f R R and g R-R be functions that are continuous on1,1 and differentiable on (1,1). Suppose that f(-1-f(1) and 9(-1). Show that there exists c e (1,1) such that
2. Let f R R and g R-R be functions that are continuous on1,1 and differentiable on (1,1). Suppose that f(-1-f(1) and 9(-1). Show that there exists c e (1,1) such that
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1) Show that the inverse function f -1 exists. (2) Prove that f is an open map (in the relative topology on I) (3) Prove that f1 is continuous
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1)...
Let f: R -R and g : R → Rbe some functions, and let x be a vector in R . Suppose that all the components off and g are directionally differentiable at x, and that g is such that, for all w RM, y +az) - g(y) y, w Then the composite function F(x)-g(f(x)) is directionally differentiable at x and the following chain rule holds: F, (x,d)=g'(f(x);f,(x,d)), YdER".
Let f: R -R and g : R → Rbe some...
3-2. Prove Theorem 3.2. Theorem 3.2 Let I S R be an open interval, xe I, and let f. 8:1\{x} → R be functions. If there is a number 8 > 0 so that f and g are equal on the subset 12 € 7\(x): 13-X1 < 8 of I\(x), then f converges at x iff g converges at x and in this case the equality lim f(x) = lim g(z) holds.
B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Define dsup(f,g)-sup |f(t)-g(t) and di(f.g)f (t)- g(t)ldt (f,g E C) tEI (i) Prove that dsup is a metric on C (ii) Prove that di is a metric on C. (You may use any standard properties of continuous functions and integrals, provided you make your reasoning clear.) 6 i) Let 1 denote the constant function on I with value 1. Give an explicit...
(2) Suppose that f and 9 are differentiable on an open interval I and that a € R either belongs to I or is an endpoint of I. Suppose further that g and g' are never zero on I\{a} and that lim f(x) is of the form 0/0. (a) If there is an M ER such that f'(2)/'(x) < M for all x E I\{a}, prove that \$(r)/g(x) < M for all x € I\{a}. (b) Is this result true...
Let f be defined on an open interval I containing a point a (1) Prove that if f is differentiable on I and f"(a) exists, then lim h-+0 (a 2 h2 (2) Prove that if f is continuous at a and there exist constants α and β such that the limit L := lim h2 exists, then f(a)-α and f'(a)-β. Does f"(a) exist and equal to 2L?
Let f be defined on an open interval I containing a point a...
C(10.1]) be the set of continuous functions f : lo. 11 → R 5) Let R from the interval [0, 1] to the real numbers. For any number ce [0, 1] (a) Show that the set R is a ring and that the set Ic is an ideal of R. (b) Is I UI2 and ideal? Is I, nI an ideal?
C(10.1]) be the set of continuous functions f : lo. 11 → R 5) Let R from the interval...