2. Consider f(x)={ x2 sin (1) xメ0 x) = (a) Show the function has a derivative...
Question 2-Part B: How many inflection points for the function whose second derivative is f"(x) sin(3x)-cos(x2) for 0 < x < 3 Question 2-Part B: How many inflection points for the function whose second derivative is f"(x) sin(3x)-cos(x2) for 0
Let f(z) e-1/2.2 for xメ0, f(0) = 0. (a) Show that the derivative fk (0) exists for all k 21. So, f is Coo everywhere on R. b) Show that the Taylor series of f about p -0 converges everywhere on R but that it represents f only at the origin.
7. Consider the function f:R + R defined by f(x) = x < 0, 3 > 0. e-1/x2, Prove that f is differentiable of all orders and that f(n)(0) = 0 for all n e N. Conclude that f does not have a convergent power series expansion En Anx" for x near the origin. [We will see later in this class that this is impossible for holomorphic functions, namely being (complex) differentiable implies that there is always a convergent power...
Compute a FD second order approximation of the first derivative of the function f(x) = sin(x2) at x = 1.5 using x = 0.1
6. Show F,{f"(x)}=-Ff()}+0f(0). F{f(x)} = f (x) sin ox dr and f"(x) is the second derivative of fx) with respect to x (5%), and write down the assumptions that f(x), f'(x) and f"(x) have to satisfy. (5%) 6. Show F,{f"(x)}=-Ff()}+0f(0). F{f(x)} = f (x) sin ox dr and f"(x) is the second derivative of fx) with respect to x (5%), and write down the assumptions that f(x), f'(x) and f"(x) have to satisfy. (5%)
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...
*PLEASE DO IN MATHEMATICA* {:1, ifr+ 13. Consider the function f(x)- nction,f(x)-e-r/rifx#0 a. Plot the graph of this function using Mathematica. b. Use the limit definition of the derivative and LHopital's Rule to show that every higher-order derivative of f at r 0 vanishes. c. Find the MacLaurin series for f. Does the series converge to f? {:1, ifr+ 13. Consider the function f(x)- nction,f(x)-e-r/rifx#0 a. Plot the graph of this function using Mathematica. b. Use the limit definition of...
1. Consider the function -F5 sin(r) for r f(x) =2 for 1< 3 2-25 for 3 x2 -9x + 20 Evaluate the following limits You do not have to cite limit laws, but you must show how you arrived at your answer If a limit Does Not Exist, explain why. You should use oo or -oo where applicable Calculating the limit using L'Hopital's Rule will receive NO CREDIT. (a) lim f(x) r-+0 (b) lim f(x)= z-1 (e) lim f(z) (d)...
Question l: Consider the function f(x) = sin(parcsinx),-1 < x < 1 and p E R (a) Calculate f(0) in terms of p. Simplify your answer completely fX) sin(p arcsinx) f(o) P The function fand its derivatives satisfy the equation where f(x) denotes the rth derivative of f(x) and f (b) Show thatf0(n2p2)f(m)(o) (x) is f(x). (nt2) (nti) (I-x) (nt 2 e 0 (c) For p E R-仕1, ±3), find the MacLaurin Series for f(x), up to and including the...
Fx 0. Show that =-- dx Fy dy 8. Suppose y is a function of z, F(x, y) = 0, and F,メO. Show that dr--Fr 9. Fid the critical points of f(z, y) if any exist, for (a, y) = ex sin y 10. Calculate the iterated integral: ysin(zy)d dy Fx 0. Show that =-- dx Fy dy 8. Suppose y is a function of z, F(x, y) = 0, and F,メO. Show that dr--Fr 9. Fid the critical points...