p380 Section 5.8: Inverse Trigonometric Functions: Differentiation
Definition of Inverse Trigonometric Functions ­ We are only going to deal with arcsin, arccos, and arctan (not arccsc, arcsec and arccot)
Function y = arcsin x iff sin y = x
y = acrcos x iff cos y = x
y = arctan x iff tan y = x
Domain
­1 ≤ x ≤ 1
­1 ≤ x ≤ 1
­∞ ≤ x ≤ ∞
Range
­π/2 ≤ y ≤ π/2
0 ≤ y ≤ π
­π/2 ≤ y ≤ π/2
y = arcsin x
y = arccos x
y = arctan x
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Example 1: Evaluating Inverse Trigonometric Functions
Evaluate each of the following:
(a).
(b).
(c).
(d).
** Remember that inverse functions have the properties
and
Properties of Inverse Trigonometric Functions
If ­1 ≤ x ≤ 1 and ­π/2 ≤ y ≤ π/2, then sin(arcsin x) = x and arcsin(sin y) = y
If ­π/2 ≤ y ≤ π/2, then tan(arctan x) = x and arctan(tan y) = y
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Example 2: Solving an Equation
Example 3: Using Right Triangles
(a). Given y = arcsin x, where 0 < y < π/2, find cos y
1
y
x
Because y = arcsin x, you know that sin y = x. This relationship between x and y can be represented by a right triangle.
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Theorem 5.18: Derivatives of Inverse Trigonometric Functions
Let u be a differentiable function of x
Example 4: Differentiating Inverse Trigonometric Functions
(a).
(b).
(c).
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Example 5: A Derivative That Can Be Simplified
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Example 6: Analyzing an Inverse Trigonometric Graph
Analyze the graph of Critical Number is x = 0
Now we can use the First Derivative Test to find the extrema
­∞ ≤ x ≤ 0
0 ≤ x ≤ ∞
­
+
Decreasing
Increasing
Since the first derivative intervals change from decreasing to increasing at x = 0, a relative minimum exists at that point
Points of Inflection: y = π2/4
y = (arctan x)2
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∴The graph has a horizontal asymptote at y = π /4
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Algebraic Functions Polynomial Functions
Rational Functions
Functions involving Radicals
vs Transcendental Functions
Logarithmic Functions
Exponential Functions
Trigonometric Functions
Inverse Trig Functions
Basic Differentiation Rules for Elementary Functions
(1).
(2).
(4).
(3).
(5).
(7).
(6).
(8).
(10).
(9).
(12).
(11).
(13).
(14).
(15).
(16).
(17).
(19).
(18).
(20).
(21).
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HW p386 #8, 12, 14, 18, 32, 42, 48, 52, 54, 56, 64
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