4) (5 marks) Find a smooth function f(x) for x є [-1, 1] that cannot be...
4. Consider the following partial information about a function f(x): S.x2, 0<x<I, (2-x), 1<x<2. Given that the function can be extended and modelled as a Fourier cosine-series: (a) Sketch this extended function in the interval that satisfies: x <4 (b) State the minimum period of this extended function. (C) The general Fourier series is defined as follows: [1 marks] [1 marks] F(x) = 4 + ] Ancos ("E") + ] B, sin("E") [1 marks] State the value of L. (d)...
4. Let the function f(x) = "be 1-1 for constants a and b. No work, no credit. a) (10 pts) Find the inverse f-'(x). b) (10 pts) Use part (a) to find the range of f(x). No graphs. Explain your answer. Give answer in interval notation. c) (10 pts) Find the range of f interval notation (x). No graphs. Explain your answer. Give answer in
1+1 5. [10 marks) Find all values of k such that f(x) = -r2 + 2kr +1" has domain (-0,0). 6. [10 marks] Sketch the graph of y = -2 -2 log: (2-2) and find its domain, range, -intercept and y-intercept. Show your transformations step by step. 7. (12 marks] (a) [8 marks) Find the equation of the tangent line to the graph of the function f(x) = 3.0" - 6x2 - 7 at = 2 (b) (4 marks) Find...
3) [8 marks] Find fog and go f, where fx) -x+2 and glx) x+3, are functions from R to R. 4) [9 marks] Consider the proposed function,f: A x B x C ? D, as defined in the following table, where A B-C-D 0,1 Input Output ABC? D 1 0I 1 00 0 0 0 0I 0 0 Is fa function? Justify your answer. That is, if it is a function use the arrow diagram to show it is a...
BONUS QUESTION (2 MARKS): A rational function is defined by where m, n є NU0} such that not every coefficient qi İs 0 and f(z) is not defined at the values of r for which q(r) = 0. What 'special' choice(s) of q(r) will give the result that the set of all polynomial functions, p(x) P +Pn-1-1 form a subset of the rational functions (as defined above)?
2. Let 6 marks (a) Find f(x),f"(x), and f"(x). (b) Find the second order Taylor expansion of f at 1, namely f(r) = ao + ala-1 ) + a2(z-1)2 + R2(x), where Ra is the remainder. You should find ao, a, a2, and R(p). 8 marks that the error in this estimation (i.e., R2(0.9)1) is at most 10-3. 6 marks (c) Use the Taylor expansion found above to estimate the value of f(0.9). Show Find f(x), f"(), and f" (b)...
5. Let f(x)- arctan(x) (a) (3 marks) Find the Taylor series about a 0 for f(x). Hint: - arctan(x) - dx You may assume that the Taylor series for f(x) converges to f (x) for values of x in the interval of convergence (b) (3 marks) What is the radius of convergence of the Taylor series for f(x)? Show that the Taylor series converges at x-1. (c) (3 marks) Hence, write T as a series (d) (3 marks) Go to...
Find the domain and the range of the function. 4 f(x)=1-7x+ 5, for x < 7. for x 27 What is the domain of the given function? (Type your answer in interval notation.) What is the range of the graph of a function? (Type your answer in interval notation.) Enter your answer in each of the answer boxes
Problem #10: [2 marks] Find the two critical values of the function f(x) = x2/5 (6 – 3x). Problem #10: separate your answers with a comma
Given the function f(x) = 4x +5 defined on the interval (0, 3, denote by fe the even 3,3 of f extension on Find fep, the Fourier series expansion of fe плг пте ao + 2 bsin fer (x) а, COs n-1 that is, find the coefficients ao an, and bn With n> 1 ao ат W |1 l Given the function f(x) = 4x +5 defined on the interval (0, 3, denote by fe the even 3,3 of f...