Ms. Battaglia AP Calculus
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Transcript of Ms. Battaglia AP Calculus
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5-6 Inverse Trig Functions: Differentiation
Objective: Develop properties of the 6 inverse trig functions and differentiate an inverse trig function.
Ms. BattagliaAP Calculus
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Function Domain RangeDefinitions of Inverse Trig Functions
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y = arcsinx y = arccosx
Graphs of Inverse Trig Functions
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y = arctanx y = arccscx
Graphs of Inverse Trig Functions
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y = arcsecx y = arccotx
Graphs of Inverse Trig Functions
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a. b.
c. d.
Evaluating Inverse Trig Functions
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If -1 < x < 1 and –π/2 < y < π/2 then
sin(arcsinx) = x and arcsin(siny) = y
If –π/2 < y < π/2, then
tan(arctanx) = x and arctan(tany) = y
If |x| > 1 and 0 < y < π/2 or π/2 < y < π, then
Sec(arcsecx) = x and arcsec(secy) = y.
Similar properties hold for other inverse trig functions.
Properties of Inverse Trig Functions
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arctan(2x – 3) = π/4Solving an Equation
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a. Given y = arcsinx, where 0 < y < π/2, find cos y.b. Given y = arcsec( ), find tan y.
Using Right Triangles
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Let u be a differentiable function of x.
Derivatives of Inverse Trig Functions
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a.
b.
c.
d.
Differentiating Inverse Trig Functions
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A Derivative That Can Be Simplified
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A photographer is taking a picture of a painting hung in an art gallery. The height of the painting is 4 ft. The camera lens is 1 ft below the lower edge of the painting. How far should the camera be from the painting to maximize the angle subtended by the camera lens?
Maximizing an Angle
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See Page 378 for a Review of Basic Differentiation Rules for Elementary Functions.
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AB: Read 5.6 Page 379 #5-11 odd, 17, 27, 29, 43-51 odd
BC: AP Sample
Classwork/Homework