calculus

posted by .

4. Given the function f defined by f(x) = cos2x for -π≤ x ≤π

a. Find the x-intercepts of the graph of f.
b. Find the x and y coordinates of all relative maximum points of f. Justify your answer.
c. Find the intervals on which the graph of f is increasing.

  • calculus -

    Make a rough sketch of y = cos 2x

    x-intercepts, when cos 2x = 0
    2x = π/2 or 2x = 3π/2
    x = π/4 or x = 3π/4
    period of cos2x is π, thus adding/subracting π to any answer will yield a new answer.
    π/4 - π = -3π/4
    3π/4 - π = -π/4

    x-intercepts are ±π/4, ±3π/4

    f'(x) = 2sin 2x
    = 0 for max/min
    2sin 2x = 0
    sin 2x= 0
    2x = 0 , π , 2π
    x = 0, π/2, π , again, since the period is π other answers are
    x = -π/2, 0, -π
    A quick look at your y = cos 2x will show you which of these are maximums and which are minimum
    Max: at x = -π, 0, π
    max points: (-π,2), (0,2), and (π,2)

    To show where y = cos 2x is increasing,
    f'(x) has to be positive
    f'(x) = -2sin 2x has to be positive
    so take a look where y = -2sin 2x lies above the x-axis
    or
    Since you have the sketch of y = cos 2x, it is easy to see that the curve is increasing between -π/2 and 0 and again between π/2 and π

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. calculus

    In spherical coordinates, the inequalities 0≤ρ≤2, 3π/4 ≤ φ ≤ π 0 ≤ θ ≤ 2π, describe what kind of shape?
  2. Pre Calculus

    sec x = -2 , π ≤ x ≤ 3π Use a calculator to solve for x in the given interval 9)sinθ=0.1473, 0°≤θ≤90° A) -8.47° B) 8.38° C) 98.47° D) 8.47° 10) cosθ=0.7563,0°≤θ≤90° …
  3. Calculus please help

    Determine the points of inflection of the function. f(x) = x + sin x (−2π ≤ x ≤ 2π)
  4. Calculus

    Determine the points of inflection of the function. f(x) = x + sin x (−2π ≤ x ≤ 2π) I'm still completely confused
  5. calculus

    3. Given the function defined by y = x + sinx for all x such that -π/2<=x<=3π/2 a. Find the coordinate of all maximum and minimum points on the given interval. Justify your answers. b. Find the coordinates of all points …
  6. calculus

    A particle moves on the x-axis so that its velocity at any time t is given by v(t) = sin 2t. At t = 0, the particle is at the origin. a)For 0 ≤ t ≤ π, find all values of t for which the particle is moving to the left. …
  7. PRE - CALCULUS

    Eliminate the parameter t. Find a rectangular equation for the plane curve defined by the parametric equations. x = 6 cos t, y = 6 sin t; 0 ≤ t ≤ 2π A. x2 - y2 = 6; -6 ≤ x ≤ 6 B. x2 - y2 = 36; -6 ≤ …
  8. Calculus 1

    Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (If an answer does not exist, enter DNE.) f(t) = 6 cos t, −3π/2 ≤ t ≤ 3π/2
  9. pre-calculus

    y=tanx -2π≤x≤2π y=cotx -2π≤x≤2π y=cscx -2π≤x≤2π y=secx -2π≤x≤2π
  10. Calc 12

    Let f be the function defined by f(x) = (x^2 + 1)e^-x for -4≤x≤4. a. For what value of x does f reach its absolute maximum?

More Similar Questions