Consider the function f(x) = 1- (x - 1) defined on the interval 0
Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
Can you help me with graphing question 1b?
![1. Consider the function f(x) =1- (x - 1)? defined on the interval 0 < x < 2.
(a) Derive a general expression for the coefficients in the Fourier Cosine series for f(x). Then write
out the Fourier series through the first four nonzero terms.
(b) Graph the extension of f(x) on the interval (-6, 6) that represents the pointwise convergence of
the Cosine series in (a). At jump discontinuities, identify the value to which the series converges.
2. Consider the function f(x) defined on 0 < x < 2 (see graph).
(a) Derive a general expression for the coefficients in the
Fourier Sine series, f(x) - b, sin(nax/2).
(b) Graph the extension of f(x) on the interval (-6, 6) that
represents the pointwise convergence of the Sine series.
At jump discontinuities, identify the value to which the
series converges.
See page 2 for Problem 3.
3. Linear Oscillator with Periodic Forcing.
Consider the second-order differen-
-
-
tial equation, L[y] = y" + 2y = f(t),
where f(t) is the periodic square
wave shown in the figure.
-3x
3x
The Fourier trigonometric series for f(t) is given by,
sin(3t), sin(5t)
fC) -+E sin(2k – 1)) = + (sin() +
+= sin(t) +
3
f(t)
5
(a) Derive a Fourier series representation for a particular solution to L[y] = f(t), as follows:
• For k = 0, 1,2,.., find a particular solution y(t) satisfying y" + 2yk = fi(t), where fa(t)
is the kth term in the Fourier series for f(t).
• Sum over all yk(t) to get the solution yp(t).
(b) Write out the first four terms in the Fourier series for yp(t).
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