MA 123 (Calculus I) Lecture 16: October 12th, 2016 Section A3

Professor Joana Amorim, jamorim@bu.edu

What is on today

1 Derivatives of Inverse Trig Functions- Wrap-up 11.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1 Derivatives of Inverse Trig Functions- Wrap-up

Briggs-Cochran-Gillett §3.10 pp. 214 – 223

We have seen at the last lecture:

Example 1 (§3.10 Ex. 24, 28, 34, 36, 68). Evaluate the derivatives of the following functions.

1. f(w) = sin(sec−1(2w))

2. f(x) = sin(tan−1(lnx))

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MA 123 (Calculus I) Lecture 16: October 12th, 2016 Section A3

Example 2 (§3.10 Ex. 34). Find an equation of the line tangent to the graph of f(x) =sec−1(ex) at the point

(ln 2, π

3

).

1.1 Applications

Example 3 (§3.10 Ex. 36). A small plane, moving at 70 m/s, flies horizontally on a line400 m directly above an observer. Let θ be the angle of elevation of the plane (see figure).

a. What is the rate of change of the angle of elevation dθdx

when the plane is x = 500 mpast the observer?

b. Graph dθdx

as a function of x and determine the point at which θ changes most rapidly.

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MA 123 (Calculus I) Lecture 16: October 12th, 2016 Section A3

Example 4 (§3.10 Ex. 68). A biologist standing at the bottom of an 80 − ft vertical cliffwatches a peregrine falcon dive from the top of the cliff at a 45◦ angle from the horizontal(see figure).

a. Express the angle of elevation θ from the biologist to the falcon as a function of thehight h of the bird above the ground. ( Hint: The vertical distance between the top ofthe cliff and the falcon is 80− h).

b. What is the rate of change of θ with respect to the bird’s height when it is 60 ft abovethe ground?

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