True or False:
If f(x) and g(x) are two differentiable functions on an interval (a,b), and f(x)>g(x) on (a,b), then f'(x)>g'(x).
True or False: If f(x) and g(x) are two differentiable functions on an interval (a,b), and...
6. Suppose that f and g are differentiable functions on an interval (a, b), and suppose that for all (a, b) we have f'(z) = g(z) and g'(z) = -f(x). Show that f2+g2 is constant.
11. Circle true or false. No justification is needed. (14 points) (a) If f(x) - o(g(x), and both functions are continuous and positive, then fix dz converges. TRUE FALSE (b) If f(x)- o(g(x)), then f(x)gx)~g(x). TRUE FALSE (c) If the power series Σ an(x + 2)" converges atェ= 5, then it must km0 converge at =-6. TRUE FALSE (d) There exists a power series Σ akz" which converges to f(z)-I on some interval of positive length around FALSE TRUE (e)...
Recall: Given two functions f(t) and g(t), which are differentiable on an interval I, • If the Wronskian W(8,9)(to) #0 for some to E I, then f and g are linearly independent for all te I. • If f(t) and g(t) are linearly dependent on I, then W (8,9)(t) = 0 for allt € 1. Note: This does NOT say that "If W(8,9)(x) = 0, then f(x) and g(2) are linearly dependent. Problem 2 Determine if the following functions are...
Suppose that fn(x) converges to f(x) uniformly, that the functions fn(x) are all differentiable, and that the function f(x) is also differentiable. (All of these conditions are assumed to be true on a bounded, closed interval [a, b].) Prove or disprove: lim as n goes to infinity fn'(x) = f'(x)
1,2 true or false If f :(a,b) + R is differentiable, then [f] : (a,b) + R is also differentiable. The series Σ, 1000 n=1 (î.01)n converges absolutely.
please help!! If the graphs of two differentiable functions f(x) and g(x) start at the same point in the plane and the functions have the same rate of change at every point, do the graphs have to be identical? Give reasons for your answer A corollary of the Mean Value Theorem states that if f7x): g7x) at each point x in an open interval (a,b), then there exists a constant C such that f(x)= g(x)-C for all Xe(a,b). That is,...
True or False If f is differentiable everywhere and f^′(x)<0 for all x, then lim x→∞ f(x)= −∞
Problem 1: Determine whether the statement is true or false. If the statement is true, then prove it. Otherwise, provide a counterexample. (a) If a continuous function f:R +R is bounded, then f'(2) exists for all x. (b) Suppose f.g are two functions on an interval (a, b). If both f + g and f - g are differentiable on (a, b), then both f and g are differentiable on (a,b). Problem 2: Define functions f,g: RR by: x sin(-),...
(a) Can there be differentiable functions f,g (on R) with g(0)-f(0) 0 and f()g(x) for all z E R? What about if we ask (only) for continuous functions f,g? (a) Can there be differentiable functions f,g (on R) with g(0)-f(0) 0 and f()g(x) for all z E R? What about if we ask (only) for continuous functions f,g?
1. Determine whether the statement is true or false. If false, explain why and correct the statement (T/FIf)exists, then lim ()f) o( T / F ) If f is continuous, then lim f(x) = f(r) (TFo)-L, then lim f(x)- lim F(x) "( T / F ) If lim -f(x)s lim. f(x) L, then lim f(x)s 1. "(T/F) lim. In x -oo . (T/F) lim0 ·(T / F ) The derivative f' (a) is the instantaneous rate of change of y...