Question

52. (a) Sketch an accurate graph of the cosine on the domain (-2?, 2?). Use this to sketch an accurate graph of the secant function on the same set of axes. (b) Identify at least three domains of length ? where the secant is one-to-one. 53. (a) Sketch y = sec x on a restricted domain of length ? where the function is one-to-one. Choose such a domain with angles in Quadrants 1 and 3 only. (b) Sketch a graph of the inverse secant function defined using this domain restriction. (c) Give the domain and range of the inverse secant according to this definition. (d) Give the open intervals where the inverse secant is increasing or decreasing. What does this tell you about the derivative of the inverse secant? (e) Show that dy/dx = 1 / (x sqrt{x^2 - 1}). 54. (a) Sketch y = sec x on a restricted domain of length ? where the function is one-to-one. Choose such a domain with angles in Quadrants 1 and 2 only. (b) Sketch a graph of the inverse secant function defined using this domain restriction. Give the domain and range of the inverse secant according to this definition. (c) Give the open intervals where the arcsecant is increasing or decreasing. What does this tell you about the derivative of the inverse secant? (d) Show that dy/dx = 1 / (|x| sqrt{x^2 - 1}). Theorem 11 (Derivative of the Inverse Secant) frac{d}{dx} sec^{-1} x = 1 / (|x| sqrt{x^2 - 1})

          52. (a) Sketch an accurate graph of the cosine on the domain (-2?, 2?). Use this to sketch an accurate graph of the secant function on the same set of axes.
(b) Identify at least three domains of length ? where the secant is one-to-one.
53. (a) Sketch y = sec x on a restricted domain of length ? where the function is one-to-one. Choose such a domain with angles in Quadrants 1 and 3 only.
(b) Sketch a graph of the inverse secant function defined using this domain restriction.
(c) Give the domain and range of the inverse secant according to this definition.
(d) Give the open intervals where the inverse secant is increasing or decreasing. What does this tell you about the derivative of the inverse secant?
(e) Show that dy/dx = 1 / (x sqrt{x^2 - 1}).
54. (a) Sketch y = sec x on a restricted domain of length ? where the function is one-to-one. Choose such a domain with angles in Quadrants 1 and 2 only.
(b) Sketch a graph of the inverse secant function defined using this domain restriction. Give the domain and range of the inverse secant according to this definition.

(c) Give the open intervals where the arcsecant is increasing or decreasing. What does this tell you about the derivative of the inverse secant?
(d) Show that dy/dx = 1 / (|x| sqrt{x^2 - 1}).

Theorem 11 (Derivative of the Inverse Secant)
frac{d}{dx} sec^{-1} x = 1 / (|x| sqrt{x^2 - 1})

$52. (a) Sketch an accurate graph of the cosine on the domain (-2?, 2?). Use this to sketch an accurate graph of the secant function on the same set of axes. (b) Identify at least three domains of length ? where the secant is one-to-one. 53. (a) Sketch y = sec x on a restricted domain of length ? where the function is one-to-one. Choose such a domain with angles in Quadrants 1 and 3 only. (b) Sketch a graph of the inverse secant function defined using this domain restriction. (c) Give the domain and range of the inverse secant according to this definition. (d) Give the open intervals where the inverse secant is increasing or decreasing. What does this tell you about the derivative of the inverse secant? (e) Show that dy/dx = 1 / (x sqrtx^2 - 1). 54. (a) Sketch y = sec x on a restricted domain of length ? where the function is one-to-one. Choose such a domain with angles in Quadrants 1 and 2 only. (b) Sketch a graph of the inverse secant function defined using this domain restriction. Give the domain and range of the inverse secant according to this definition. (c) Give the open intervals where the arcsecant is increasing or decreasing. What does this tell you about the derivative of the inverse secant? (d) Show that dy/dx = 1 / (|x| sqrtx^2 - 1). Theorem 11 (Derivative of the Inverse Secant) fracddx sec^-1 x = 1 / (|x| sqrtx^2 - 1)$

Added by Jonathan M.

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