Solved: In 110 determine whether the function is even

In Problems 110 determine whether the function is even, odd, or neither.f(x) sin 3x

In Problems 110 determine whether the function is even, odd, or neither.f(x) x cos x

In Problems 110 determine whether the function is even, odd, or neither.f(x) x 2  x

In Problems 110 determine whether the function is even, odd, or neither.f(x) x3  4x

In Problems 110 determine whether the function is even, odd, or neither.f(x) e x

In Problems 110 determine whether the function is even, odd, or neither.f(x) ex  ex

In Problems 110 determine whether the function is even, odd, or neither.f(x)  x2 , x2 , 1  x  0 0  x  1

In Problems 110 determine whether the function is even, odd, or neither.f(x)  x  5, x  5, 2  x  0 0  x  2

In Problems 110 determine whether the function is even, odd, or neither.f(x) x3 , 0  x  2

In Problems 110 determine whether the function is even, odd, or neither.f(x) x5

In Problems 1124 expand the given function in an appropriate cosine or sine series.f(x)  1, 1,   x  0 0  x 

In Problems 1124 expand the given function in an appropriate cosine or sine series.f(x)  1, 0, 1, 2  x  1 1  x  1 1  x  2

In Problems 1124 expand the given function in an appropriate cosine or sine series.f(x) x ,p  x  p

In Problems 1124 expand the given function in an appropriate cosine or sine series.f(x) x, p  x  p

In Problems 1124 expand the given function in an appropriate cosine or sine series.f(x) x2, 1  x  1

In Problems 1124 expand the given function in an appropriate cosine or sine series.f(x) x x , 1  x  1

In Problems 1124 expand the given function in an appropriate cosine or sine series.f(x) p2  x2, p  x  p

In Problems 1124 expand the given function in an appropriate cosine or sine series.f(x) x3, p  x  p

In Problems 1124 expand the given function in an appropriate cosine or sine series.f(x)  x  1, x  1,   x  0 0  x 

In Problems 1124 expand the given function in an appropriate cosine or sine series.f(x)  x  1, x  1, 1  x  0 0  x  1

In Problems 1124 expand the given function in an appropriate cosine or sine series.f(x)  1, x, x, 1, 2  x  1 1  x  0 0  x  1 1  x  2

In Problems 1124 expand the given function in an appropriate cosine or sine series.f(x)   , x, , 2  x     x   x  2

In Problems 1124 expand the given function in an appropriate cosine or sine series.f(x) sin x ,p  x  p

In Problems 1124 expand the given function in an appropriate cosine or sine series.f(x) cos x, p2  x  p2

In Problems 2534 find the half-range cosine and sine expansions of the given function.f(x)  1, 0, 0  x  1 2 1 2  x  1

In Problems 2534 find the half-range cosine and sine expansions of the given function.f(x)  0, 1, 0  x  1 2 1 2  x  1

In Problems 2534 find the half-range cosine and sine expansions of the given function.f(x) cos x, 0  x  p2

In Problems 2534 find the half-range cosine and sine expansions of the given function.f(x) sin x, 0  x  p

In Problems 2534 find the half-range cosine and sine expansions of the given function.f(x)  x,  x, 0  x  >2 >2  x 

In Problems 2534 find the half-range cosine and sine expansions of the given function.f(x)  0, x  , 0  x   x  2

In Problems 2534 find the half-range cosine and sine expansions of the given function.f(x)  x, 1, 0  x  1 1  x  2

In Problems 2534 find the half-range cosine and sine expansions of the given function.f(x)  1, 2  x, 0  x  1 1  x  2

In Problems 2534 find the half-range cosine and sine expansions of the given function.f(x) x2  x, 0  x  1

In Problems 2534 find the half-range cosine and sine expansions of the given function.f(x) x(2  x), 0  x  2

In Problems 3538 expand the given function in a Fourier series. f(x) x2, 0  x  2p

In Problems 3538 expand the given function in a Fourier series. f(x) x, 0  x  p

In Problems 3538 expand the given function in a Fourier series.f(x) x  1, 0  x  1

In Problems 3538 expand the given function in a Fourier series. f(x) 2  x, 0  x  2

In Problems 39 and 40 proceed as in Example 4 to find a particular solution xp(t) of equation (11) when m 1, k 10, and the driving force f(t) is as given. Assume that when f(t) is extended to the negative t-axis in a periodic manner, the resulting function is odd.f(t)  5, 5, 0  t   t  2 ; f(t  2 ) f(t)

In Problems 39 and 40 proceed as in Example 4 to find a particular solution xp(t) of equation (11) when m 1, k 10, and the driving force f(t) is as given. Assume that when f(t) is extended to the negative t-axis in a periodic manner, the resulting function is odd.f(t) 1  t, 0  t  2; f(t  2) f(t)

In Problems 41 and 42 proceed as in Example 4 to find a particular solution xp(t) of equation (11) when and the driving force f(t) is as given. Assume that when f(t) is extended to the negative t-axis in a periodic manner, the resulting function is even. f(t) 2pt  t 2 , 0  t  2p; f(t  2p) f(t)

In Problems 41 and 42 proceed as in Example 4 to find a particular solution xp(t) of equation (11) when and the driving force f(t) is as given. Assume that when f(t) is extended to the negative t-axis in a periodic manner, the resulting function is even. f(t)  t, 1  t, 0  t  1 2 1 2  t  1 ; f(t  1) f(t)

(a) Solve the differential equation in Problem 39, x  10x f(t), subject to the initial conditions x(0) 0, x(0) 0. (b) Use a CAS to plot the graph of the solution x(t) in part (a). 4

(a) Solve the differential equation in Problem 41, subject to the initial conditions x(0) 1, x(0) 0. (b) Use a CAS to plot the graph of the solution x(t) in part (a).

Suppose a uniform beam of length L is simply supported at x 0 and at x L. If the load per unit length is given by w(x) w0 x L, 0  x  L, then the differential equation for the deflection y(x) is where E, I, and w0 are constants. (See (4) in Section 5.2.) (a) Expand w(x) in a half-range sine series. (b) Use the method of Example 4 to find a particular solution yp(x) of the differential equation. 4

Proceed as in Problem 45 to find a particular solution yp(x) when the load per unit length is as given in Figure 11.3.13. EI d4y dx4 w0 x L , x w(x) L/3 2L/3 L w0 FIGURE 11.3.13 Graph for Problem 46

When a uniform beam is supported by an elastic foundation and subject to a load per unit length w(x), the differential equation for its deflection y(x) is where k is the modulus of the foundation. Suppose that the beam and elastic foundation are infinite in length (that is,   x  ) and that the load per unit length is the periodic function Use the method of Example 4 to find a particular solution yp(x) of the differential equation.

Prove properties (a), (c), (d), (f), and (g) in Theorem 11.3.1.

There is only one function that is both even and odd. What is it?

As we know from Chapter 4, the general solution of the differential equation in Problem 47 is y yc  yp. Discuss why we can argue on physical grounds that the solution of Problem 47 is simply yp. [Hint: Consider y yc  yp as ] x

In Problems 51 and 52 use a CAS to plot graphs of partial sums {SN(x)} of the given trigonometric series. Experiment with different values of N and graphs on different intervals of the x-axis. Use your graphs to conjecture a closed-form expression for a function f defined for 0  x  L that is represented by the series.f(x)  4   n1  (1)n  1 n2 cos nx

In Problems 51 and 52 use a CAS to plot graphs of partial sums {SN(x)} of the given trigonometric series. Experiment with different values of N and graphs on different intervals of the x-axis. Use your graphs to conjecture a closed-form expression for a function f defined for 0  x  L that is represented by the series.f(x) 1 4  4 2  n1 1 n2 1  cos n 2  cos n 2 x

Is your answer in Problem 51 or in Problem 52 unique? Give a function f defined on a symmetric interval about the origin (a, a) that has the same trigonometric series (a) as in Problem 51, (b) as in Problem 52.