(x) = f 0 (f 1 (x)) to check your answers, and then to calculate the derivatives of the other inverse trig functions: sin(arccos(x)

Transcription

1 Reviewing the worksheet On the worksheet, you learne how to compute the erivatives of inverse functions, such as ln(x), arcsin(x), etc. You use implicit i erentiation to calculate the erivatives. x arcsin(x)= cos(arcsin(x)) 2. x arctan(x)= sec 2 (arctan(x)) an the rule x f (x) = f 0 (f (x)) to check your answers, an then to calculate the erivatives of the other inverse trig functions:. x arccos(x) = 2. x arccot(x)= sin(arccos(x) csc 2 (arccot(x)) 3. x arcsec(x)= 4. x arccsc(x)= sec(arcsec(x)) tan(arcsec(x)) csc(arccsc(x)) cot(arccsc(x)) Using implicit i erentiation to calculate x arcsin(x) Take x Right han sie: So If y = arcsin(x) then x =sin(y). of both sies of x =sin(y): Left han sie: x x = y sin(y) = cos(y) x x y x = cos(arcsin(x)). = cos(arcsin(x)) y x

2 Simplifying cos(arcsin(x)) Call arcsin(x) =. sin( ) =x θ Key: This is a simple triangle to write own whose angle has sin( ) =x Calculating x arctan(x). We foun that x arctan(x) = sec 2 (arctan(x)) = Simplify this expression using sec(arctan(x)) 2 y x = arctan(x) 2 = sec(arctan(x)) +x 2

3 To simplify the rest, use the triangles arccos(x) arcsec(x) arccsc(x) arccot(x)

4 Toay: Relate rates. Example: Suppose you has a 5m laer resting against a wall.??? 5m 2 mps Move the base out at 2 m/s How fast oes the top move own the wall? Toay: Relate rates. Example: Suppose you has a 5m laer resting against a wall. y 5m x Move the base out at 2 m/s: How fast oes the top move own the wall? x t =2 y t =?? To solve, we nee to relate the variables: x 2 + y 2 =5 2 0 apple x apple 5

5 Problem: If x 2 + y 2 =5 2 for 0 apple x apple 5, an x t Di erentiate: =2,whatis y t? So 0= t 52 = x 2 + y 2 t =2x x y +2y t t =2x 2 +2 p 25 x 2 y t = 2x p 25 x 2 y t Notice: () y t < 0 (y is ecreasing) an (2) lim x5 y t = Example Suppose you have a sphere whose raius is growing at a rate of 5in/s. How fast is the volume growing when the raius is 3in? r 5 in/s Relating equation: V = 4 3 r3 Take a erivative: V t = 4 3 3r2 r t Substitute in the known values: V t r=3 = = in 3 /s

6 Take an upsie-own cone-shape bowl, with a raius of 4in at the top an a total height of 3in fill it with water at a rate of /2 in 3 /min. How fast is the height of water increasing when h=2in? r 4 h 3 Volume of a cone: V = 3 R2 H Volume of a water: V = 3 r2 h Relate r an h: r/h =4/3 so r = 4 3 h Finally, equation to i erentiate: V = 3 2 = V t = 6 27 h 3h2 t = h h = 6 27 h3 (2)2 h t h=2 So h t h=2 = 9 28 Strategy:. Fin an equation which relates the functions you nee. (a) Draw pictures (b) Sometimes you ll have to reuce the number of variables/functions to get it own to (i) the function from the rate you know, (ii) the function from the rate you want, an (iii) maybe the variable from the rate you know an want (t in the last 3 examples). 2. Take a erivative using implicit i erentiation. 3. Plug in the values you know. 4. Solve for the rate you want.

7 One more example: (from extra problems see website) 0. A boat is pulle into a ock by a rope attache to the bow of the boat a passing through a pulley on the ock that is m higher than the bow of the boat. If the rope is pulle in at a rate of m/s how fast is the boat approaching the ock when it is 8 m from the ock? On your own: (from extra problems see website) 5. A balloon which always remains spherical is being inflate by pumping in 900 cubic centimeters of gas per secon. Fin the rate at which the raius of the balloon is increasing when the raius is 5cm.. Gravel is being umpe from a conveyor belt at a rate of 30 ft 3 /min an its coarseness is such that it forms a pile in the shape of a cone whose base iameter an height are always equal. How fast is the height of the pile increasing when the pile is 0 ft high? 9. A lighthouse is on a small islan 3 km away from the nearest point P on a straight shoreline an its light turns four revolutions per minute. How fast is the beam of light moving along the shoreline when it is km from P? [hint: 4rpm means that some angle is changing at 4 2 raians per minute]