f(x) = cos(x)
on the interval [−2π, 2π]
(a) Find the x-intercepts of the graph of
y = f(x).
(Enter your answers as a comma-separated list.)
(b) Find the y-intercepts of the graph of
y = f(x).
(Enter your answers as a comma-separated list.)
(c) Find the intervals on which the graph of
y = f(x)
is increasing and the intervals on which the graph of
y = f(x)
is decreasing. (Enter your answers using interval notation.)
(d) Find the relative extrema of the graph of
y = f(x).
(Enter your answers as a comma-separated list of ordered pairs.)
−2π to 2π where f(x)=cosx
Find the exact zeros of the function
f(x) = 4x2 − 4x − 1.
(Enter your answers as a comma-separated list.)
To find the x-intercepts of the graph of f(x) = cos(x) on the interval [-2π, 2π], we need to find the values of x for which f(x) is equal to zero.
(a) The x-intercepts of the function occur when f(x) = cos(x) = 0. We know that the cosine function equals zero at multiples of π/2. So, we can find the x-intercepts by solving the equation cos(x) = 0.
Using the unit circle or knowledge of the cosine function, we know that cos(x) = 0 at x = π/2 and x = 3π/2. However, we need to find the x-intercepts within the interval [-2π, 2π].
Within this interval, the x-intercepts are x = π/2 and x = 3π/2.
Thus, the x-intercepts of the graph of f(x) = cos(x) on the interval [-2π, 2π] are (π/2, 0) and (3π/2, 0).
To find the y-intercepts of the graph of y = f(x) = cos(x), we need to find the value of y when x = 0.
(b) Substituting x = 0 into the equation y = f(x), we get y = cos(0) = 1.
Thus, the y-intercept of the graph of y = f(x) = cos(x) is (0, 1).
To find the intervals on which the graph of y = f(x) = cos(x) is increasing and decreasing, we need to analyze the derivative of the function f(x).
(c) The derivative of f(x) = cos(x) is f'(x) = -sin(x). To determine when the function is increasing or decreasing, we need to find where the derivative f'(x) is positive or negative.
The sine function is positive in the first and second quadrants, and negative in the third and fourth quadrants. So, the derivative f'(x) = -sin(x) is positive when x is in the second quadrant (π/2 to π) and the fourth quadrant (3π/2 to 2π), and negative when x is in the first quadrant (0 to π/2) and third quadrant (π to 3π/2).
Therefore, the interval on which the graph of y = f(x) = cos(x) is increasing is (π/2, π) ∪ (3π/2, 2π), and the interval on which it is decreasing is (0, π/2) ∪ (π, 3π/2).
To find the relative extrema of the graph of y = f(x) = cos(x), we need to locate the points at which the function has maximum and minimum values.
(d) Since the function f(x) = cos(x) is periodic with a period of 2π, we should focus on the interval [-2π, 2π] to locate the relative extrema.
The maximum value of f(x) = cos(x) occurs when x = 0 and x = 2π, giving us the relative maximum points (0, 1) and (2π, 1).
The minimum value of f(x) = cos(x) occurs when x = π and x = -π, giving us the relative minimum points (π, -1) and (-π, -1).
Therefore, the relative extrema of the graph of y = f(x) = cos(x) on the interval [-2π, 2π] are (0, 1), (2π, 1), (π, -1), and (-π, -1).
9 8
(a) surely you know where cos(x) = 0
(b) surely you can determine cos(0)
(c) check the graph below
(d) surely you know the min and max of cos(x)
graph at
http://www.wolframalpha.com/input/?i=cos%28x%29