(b) The Fourier transform of a piecewise smooth function f(x), for which f∞ f(x)|dx exists and is finite, is given by 1 ∞ F(k) = F[f(x)] = 2πT f(x)e-dr. -ikx ∞ i. Show that for a real valued function f(x) its Fourier transform F(k) satisfies the identity F(-k) = F(k) ii. Assuming that the function f(x) is real valued and odd (f(x) = − f(x)), show that its Fourier transform F(k) = F[f(x)] is a purely imaginary function, that is, Re(F(k)) = 0. iii. Find the Fourier transform F(k) for the function f(x) = x exp(−|x|). Hint: fre X 1 x exp(ax) dx = exp(ax) ― a exp(ax) + constant.

Find a second-degree polynomial f(x) = ax² + bx + c such that its graph has a tangent line with slope 10 at the point (2,7) and an x -intercept at (1,0).

Determine whether the pairs of the functions are linearly inde-pendent or linearly dependent on the real line. f(x) = e^x sin(2πx), g(x) = e^x cos(2πx)

On which interval is the function f(x) = x2 - 2x - 8 decreasing?  A [-2, 4]  B (-∞, 4)  C (-2, 4)  D (-∞, 1)  E (1, ∞)

Find the linearization of the function at x = 1. * f (x) 2x – 3x + 7 O y = 6 + 1(x - 3) O y = 6 + 3(x - 1) O y = 1+ 3(x - 6) O y = 3 + 6(x - 1)

For any function , With utilization of the product rule for functions of one variable (eq1), show / prove how it could be viewed as the result of the chain rule applied to a particular function of two variables(eq2). Eq 1 notation: The Product Rule If f and g are both differentiable, then (fg)' = fg' + gf' d d d [S(x)g(x)] = f(x) – [g(x)] + g(x) [f(x)] %3D dx dx dx Eed 2 The Chain Rule (Case 1) Suppose that z = f(x, y) is a differentiable func- tion of x and y, where x = g(t) and y = h(t) are both differentiable functions of t. Then z is a differentiable function of t and dz af dx of dy + dt dx dt ду di

Find f-1 if f (z) = iz + 1.

Let f(x) = Vx +7 and g(x) = x +3 a) Find (2) (2) show work. b) Give the domain of the function (-) x) in interval notation

Define F : R → R and G : R → Z by the following formulas: F(x) = (x^2)/3 and G(x) = ⌊x⌋ for every x ∈ R . (a) (G ∘ F)(2) =  (b) (G ∘ F)(−3) =  (c) (G ∘ F)(4) =

if f(x)=cospix, find the extact value of a such that f(1)-f(-1)/1-(-1)=cos a.