MAT137 Calculus! Lecture 16

Transcription

1 MAT137 Calculus! Lecture 16 Today: 4.10 Related Rated 7.1 One-to-One Functions Inverse Next: 7.7 Transcendental Functions. HOMEWORK: Watch the YouTube Videos on derivatives of exponentials and logarithms.

2 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?

3 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?

4 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.

5 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?

6 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?

7 Cinderella At midnight, Cinderella has to run away from the ball. She runs along a straight path towards the Castle s wall at a speed of 4 m/s. Noticing this, the prince asks the guards to shine a spotlight on Cinderella as she begins to run. The spotlight is located on the ground 30 m from the wall. Cinderella is 1.5 m tall and weighs 50 kg. At which rate is Cinderella s shadow on the wall changing when she is 20 m from the wall? Is the shadow increasing or decreasing?

8 Sleepy Ants Two ants are taking a nap on Cogsworth s moustache. The first one is resting at the tip of the minute hand, which is 8 cm long. The second one is resting at the tip of the hour hand, which is half the length. At what rate is the distance between the two ants changing at two o clock?

9 One-to-One Functions Definition A function f is one-to-one if it never takes on the same value twice that is, for all a and b in the domain of f, a b implies f (a) f (b) equivalently, if f (a) = f (b) implies a = b.

10 One-to-One Functions Horizontal Line Test A function is one-to-one if and only if no horizontal line intersects its graph more than once. y y y = x 3 y = x 2 One-to-One x x Not One-to-One

11 Inverse Function The inverse of f denoted f 1, is the function that reverses the effect of f. Function x y = x Apply f and then f 1 : 2 Apply f 1 and then f : 8 Apply x 3 8 Apply x 1/3 2 Inverse x y = x 1/ Apply x 1/3 2 Apply x 3 8

12 Inverse Function Definition (Inverse Function) Let f be a function with domain D and range R. If there exists a function g with domain R such that g(f(x)) = x, for all x in D f(g(x)) = x, for all x in R then g is the inverse of f and we denote it by f 1. Note 1 Do not mistake the 1 in f 1 for an exponent. f 1 1 (x) does not mean f (x). 1 To express the reciprocal f (x) using the exponent 1, we write [f (x)] 1. Note 2 If f is not one-to-one, then f 1 cannot be uniquely defined.

13 Inverse Function Theorem A function f has an inverse if and only is f is one-to-one.

14 Inverse Function Proposition In f is an invertible function with inverse f 1, then 1 Domain of (f 1 ) = Range (f ) and Range of (f 1 ) = Domain of (f ). 2 f 1 (b) = a if and only if f (a) = b.

15 Inverse Function Proposition IF f is an invertible function with inverse f 1, THEN 1 Domain of (f 1 ) = Range (f ) and Range of (f 1 ) = Domain of (f ). 2 f 1 (b) = a if and only if f (a) = b.

16 Inverse Function - Graph Proposition IF f is an invertible function with inverse f 1, THEN The graph of y = f 1 (x) is the graph of y = f (x) reflected across the line y = x. y f f 1 x y = x Graphs of f (x) = x 3 and its inverse f 1 (x) = x 1 3.

17 Inverse Function Example 1 Show that the linear function f (x) = 2x 5 is one-to-one and find its inverse. y f y f 1 f x x 5 5

18 Inverse Function In general, to find the inverse of a one-to-one function, we do the following: 1 Set y = f 1 (x). 2 Apply f to both sides to get f (y) = x. 3 Solve this equation for y in terms of x (if possible).

19 Restricting the Domain y y = x 2 y y = x 2 x x Not One-to-One One-to-One y y = x 2 y = x x