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Problem of the Day

Problem of the Day. Which of the following are antiderivatives of f(x) = sin x cos x?. I. F(x) = ½ sin2x II. F(x) = ½ cos2x III. F(x) = -¼ cos(2x). A) I only B) II only C) III only D) I and III only E) II and III only. Problem of the Day. Which of the following are antiderivatives of

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Problem of the Day

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  1. Problem of the Day Which of the following are antiderivatives of f(x) = sin x cos x? I. F(x) = ½ sin2x II. F(x) = ½ cos2x III. F(x) = -¼ cos(2x) A) I only B) II only C) III only D) I and III only E) II and III only

  2. Problem of the Day Which of the following are antiderivatives of f(x) = sin x cos x? I. F(x) = ½ sin2x II. F(x) = ½ cos2x III. F(x) = -¼ cos(2x) A) I only B) II only C) III only D) I and III only E) II and III only

  3. Inverse Trig Functions? How? None of the 6 basic trig functions has 
an inverse because they are not 1 to 1.

  4. Inverse Trig Functions? How? None of the 6 basic trig functions has 
an inverse because they are not 1 to 1. Restrict the domain and they do.  (see page 373)

  5. sin arcsin (sin y = x) (arcsin x = y) Domain [-π/2, π/2] Range [-1, 1] Domain [-1, 1] Range [-π/2, π/2] cos arccos (cos y = x) (arccos x = y) Domain [0, π] Range [-1, 1] Domain [-1, 1] Range [0, π] tan arctan (tan y = x) (arctan x = y) Domain [-∞, ∞] Range [-π/2, π/2] Domain [-π/2, π/2] Range [-∞, ∞]

  6. sec arcsec Domain |x| > 1 Range [0, π], y≠π/2 Domain [0, π], x≠π/2 Range |y| > 1 csc arccsc Domain |x| > 1 Range [-π/2, π/2], y≠0 Domain [-π/2, π/2], x≠0 Range |y| > 1 cot arccot Domain [-∞, ∞] Range [0, π] Domain [0, π] Range [-∞, ∞]

  7. Evaluate - arcsin(-½) infers that sin y = -½ in the interval [-π/2, π/2],  the angle that gives -½ as its sin is -π/6 Evaluate - arcsin(0.3) using a calculator in radian  mode y ≈ .3047

  8. Inverse Properties (see page 373) If you are in the restricted interval then tan(arctan x) = x and arctan (tan y) = y Example arctan (2x - 3) = π/4 tan(arctan (2x - 3)) = tan π/4 2x - 3 = 1   x = 2 Similar properties hold true for other trig functions.

  9. Remember the relationship between the derivative of a function and it's inverse?

  10. Remember the relationship between the derivative of a function and it's inverse? Consider the triangle 1 u

  11. 1 f(u) = sin u f '(u) = cos u du g(u) = arcsin u u

  12. Derivatives of Inverse Trig Functions

  13. Examples

  14. Find cos(arcsin x) sin = opp = x hyp 1 1. θ 1 2. a2 + b2 = c2 x2 + b2 = 12 b = x 3. cos = adj = hyp 1

  15. Find an equation for the line tangent to the graph of  y = cot-1x at x = -1

  16. Find an equation for the line tangent to the graph of  y = cot-1x at x = -1 evaluated at x = -1 gives -½ which gives you the slope of the tangent line find y when x = -1 y = cot-1(-1) = π/2 - tan-1)(-1) = π/2 - (-π/4) = 3π/4 find equation of tangent line y - 3π/4 = -½(x + 1)

  17. Conversions sec-1 x = cos-1(1/x) csc-1 x = sin-1(1/x) cot-1 x = π/2 - tan-1(x) Page 378 summarizes all rules learned so far

  18. (See page 376)

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