MAT137 Calculus! Lecture 17 Today: 4.10 Related Rated Local and Global Extrema Next: Mean Value Theorem v. 5.5-5.8 official website http://uoft.me/mat137
Arctan This inverse is called the arc tangent function: y = arctan x, x R. Since these functions are inverses to each other, we have tan(arctan x) = x for all x R; ( arctan(tan x) = x for all x π 2, π ). 2
Inverse Trigonometric Functions Calculate: 1 tan(arcsin 1 3 ) 2 arctan(tan 2) 3 tan(arctan 2)
Inverse Trigonometric Functions Calculate: sec(arctan x)
Derivative of Inverse Trigonometric Functions Find d dx arctan x.
Inverse Trigonometric Functions Name Notation Definition Maps arcsine y = arcsin x sin y = x [ 1, 1] [ π 2, ] π 2 arccosine y = arccos x cos y = x [ 1, 1] [0, π] arctangent y = arctan x tan y = x (, ) ( π 2, ) π 2 arccosecant y = arccsc x csc y = x (, 1] [1, ) [ π 2, 0) ( ] 0, π 2 arcsecant y = arcsec x sec y = x (, 1] [1, ) [ ) ( 0, π 2 π 2, π] arccotangent y = arccot x cot y = x (, ) (0, π).
Derivative of Inverse Trigonometric Functions d 1 dx (arcsin x) = 1 x 2 d 1 dx (arctan x) = x 2 +1 d dx (arcsec x) = 1 x x 2 1 d 1 dx (arccos x) = 1 x 2 d 1 dx (arccot x) = x 2 +1 d 1 dx ( arccsc x) = x. x 2 1
Finding Local and Global Extrema Example 1 Find local and global extrema of the function with this graph. 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10
Extreme Value Theorem Theorem (Extreme Value Theorem) If f is continuous on a closed and bounded interval [a, b], then f attains both a maximum value M and a minimum value m in [a, b].
Local Extrema Theorem (Local Extreme Value Theorem) If f has a local maximum or minimum at an interior point c of its domain, then f (c) = 0 or f (c) does not exist
Finding Local and Global Extrema Example 2 Find the local and global extrema of the function f (x) = x 2/3 (x 1) 3 on the interval [ 1, 2]
Finding Local and Global Extrema Example 2 Find the local and global extrema of the function f (x) = x 2/3 (x 1) 3 on the interval [ 1, 2] 1-1.0-0.5 0.5 1.0 1.5 2.0-1 -2-3
Finding Local and Global Extrema Let f be continuous on a closed and bounded interval [a, b]. The Extreme Value Theorem says that f attains a maximum and a minimum. The Local Extreme Value Theorem says the the only places where a function can possibly have an extreme value are 1 interior points where f = 0, 2 interior points where f is undefined, 3 endpoints of the domain of f.
Ladder and Slippery floor Trying to reach her calculus textbook, Belle props a 10-foot ladder against the bookcase. Unfortunately, the floor was very slippery because the Beast had just mopped it, and the base of the ladder slides away from the bookcase at a rate of 1 foot per second. How fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall?
Ladder and Slippery floor Trying to reach her calculus textbook, Belle props a 10-foot ladder against the bookcase. Unfortunately, the floor was very slippery because the Beast had just mopped it, and the base of the ladder slides away from the bookcase at a rate of 1 foot per second. According to the model we used, what happens as the top of the ladder approaches the ground? Is the model appropriate for small values of y?
Strategy 1 Read the problem carefully. 2 Draw a diagram if possible. 3 Introduce notation. Assign symbols to all quantities that are functions of time. 4 Express the given information and the (unknown) required rate in terms of derivatives. 5 Find an equation that relates the relevant variables. 6 Use the Chain Rule to differentiate both sides of the equation with respect to t. 7 Plug the given information into the resulting equation and solve for the unknown. 8 State the final answer in a coherent form, specifying the units that you are using.
UP! Thousands of balloons were tied to Carl s home, as a result the house begins to rise. The house leaves the ground 500 ft away from an observer and rises vertically at a rate of 14 ft/min. At what rate is the observer s viewing angle changing at the instant when the house is exactly 500 ft above the ground?
Shrek and Donkey At noon, after a long phone conversation fighting over where to og for lunch, Shrek and Donkey decided to go to different places. Donkey is 4 km west of Shrek s home. Shrek starts walking south at 3 km/h and, Donkey begins walking north at 5 km/h. How fast is the distance between them changing at 1:00 pm?