Find the half-range expansions of the given function. To
illustrate the convergence of the cosine and sine series, plot
several partial sums of each and comment on the graph (using
words).
f(x) = 1 if 0 < x < 1.
f(x) = π-x if 0 ≤ x ≤ π.
f(x) = x2 if 0 < x <1.
Expand in Fourier series:
Expand in fourier sine and fourier cosine series of: f(x) =
x(L-x), 0<x<L
Expand in fourier cosine series: f(x) = sinx, 0<x<pi
Expand in fourier series f(x) = 2pi*x-x^2, 0<x<2pi,
assuming that f is periodic of period 2pi, that is,
f(x+2pi)=f(x)
Find the Fourier Series for the function defined over -5 < x
< 5
f(x) = -2 when -5<x<0 and f(x) = 3 when 0<x<5
You can use either the real or complex form but must show
work.
Plot on Desmos the first 10 terms of the series along with the
original
function.
6. (a) let f : R → R be a function defined by
f(x) =
x + 4 if x ≤ 1
ax + b if 1 < x ≤ 3
3x x 8 if x > 3
Find the values of a and b that makes f(x) continuous on R. [10
marks]
(b) Find the derivative of f(x) = tann 1
1 ∞x
1 + x
. [15 marks]
(c) Find f
0
(x) using logarithmic differentiation, where f(x)...
A probability density function on R is a function f :R -> R
satisfying (i) f(x)≥0 or all x e R and (ii) \int_(-\infty )^(\infty
) f(x)dx = 1. For which value(s) of k e R is the function
f(x)= e^(-x^(2))\root(3)(k^(5)) a probability density function?
Explain.