Question

Consider the following graph of f(x). Which of the following are inflection points of f? A coordinate plane has a horizontal x-axis labeled from negative 4 to 2 in increments of 1 and a vertical y-axis labeled from negative 7 to 2 in increments of 1. From left to right, a curve falls and passes through left-parenthesis negative 3.1 comma 0 right-parenthesis to a minimum at left-parenthesis negative 2 comma negative 5 right-parenthesis. It then rises to a maximum at left-parenthesis 0 comma negative 1 right-parenthesis, and then falls steeply, passing through to left-parenthesis 1 comma negative 5 right-parenthesis. All coordinates are approximate. Select all that apply: (?1,?3) (?2,?5) (0,?1) (?3,?1) (1,?5)

Answer 1

(b) To calculate the Fourier transform of (1/3)ⁿ⁻², we'll follow a similar approach. Let's substitute the signal into the D T F T formula

X (
`e^(jw)`) = Σ (1/3)ⁿ⁻²
`e^(-jwn)`

Again, let's rewrite the summation limits to simplify the calculation:

X (
`e^(jw)`) = Σ (1/3)ⁿ⁺¹
`e^(-jwn)`

Splitting the summation into two parts

X (
`e^(jw)`) = (1/3)⁻¹ + Σ (1/3)ⁿ⁺¹
`e^(-jwn)`

X (
`e^(jw)`) = 3 + Σ (1/3)ⁿ⁺¹
`e^(-jwn)`

The first term in the equation represents a constant, and the second term represents a geometric series. Using the formula for the sum of a geometric series

X (
`e^(jw)`) = 3 + (1/3) Σ (
`e^(-jw)`)ⁿ

X (
`e^(jw)`) = 3 + (1/3) ( 1 / (1 -
`e^(-jw)`))

Simplifying further

X (
`e^(jw)`) = 3 + 1 / (3 (1 -
`e^(-jw)`))

Therefore, the of the given signal is

X (
`e^(jw)`) = 3 + 1 / (3 (1 -
`e^(-jw)`))