CHAPTER 3 DERIVATIVES (continued)

Transcription

1 CHAPTER 3 DERIVATIVES (continue) 3.3. RULES FOR DIFFERENTIATION A. The erivative of a constant is zero: [c] = 0 B. The Power Rule: [n ] = n (n-1) C. The Constant Multiple Rule: [c *f()] = c * f () D. The Sum an Difference Rule: [f() + g()] = f () + g () E. Note that we can now fin the erivative of any polynomial easily! F. The secon erivative is just the erivative of the erivative. It has many notations. G. Assignment: P. 10 {1 10} H. The Prouct Rule: [f() * g()] = f() * g () + g() * f () f ( ) g( ) * f ( ) f ( ) * g ( ) = g g I. The quotient Rule: ( ) ( ) ho [hi] minus hi [ho] over ho ho J. Notes on the quotient Rule: 1. Sometimes a quotient is really just a scalar multiple.. Sometimes a quotient can be written as a prouct. K. Higher orer erivatives an their notations. L. Higher orer erivatives on the calculator. M. Applications are easier with theorems: 1. Making piece-wise efine functions continuous an ifferentiable.. Tangent an normal lines revisite. N. Assignment: p. 10 {11, 1, 13, 14, 16, 17, 3, 4, 5, 7, an 30} 3.4. VELOCITY AND OTHER RATES OF CHANGE A. In general, the instantaneous rate of change of a function at a point a is f (a). B. Definitions: 1. Displacement is the change in position.. Motion along a line is calle rectilinear motion. C. Since the average velocity is the (change in istance)/(change in time), it is also the (isplacement)/(change in time).

2 D. Concerning motion an in particular, rectilinear motion - the (instantaneous) velocity is the erivative of the position function with respect to time. The sign of the velocity inicates irection of motion. E. Definitions: 1. Spee is the absolute value of velocity.. Acceleration is the erivative of the velocity function w.r.t. time. It measures the rate of change of the velocity w.r.t. time an is the secon erivative of the position function. F. A comparison of signs on position, velocity, an acceleration for rectilinear motion: Negative Zero Positive Position Left / below the origin groun zero / the origin right / above the origin velocity moving left / own not moving / stoppe moving right / up acceleration slowing own if v(t) > 0 constant motion speeing up if v(t) > 0 speeing up if v(t) < 0 slowing own if v(t) < 0 G. Note that an object is speeing up if v(t) an a(t) have the same sign. It is slowing own if they have the opposite signs. We will inclue the en points for speeing up or slowing own because it is a comparison, but the A.P. folks on t care if you inclue them or not. H. We will consier projectile motion now as a vertical form of rectilinear motion. I. Assignment: P. 19 {, 3, 5, an 6} J. Formulas for projectile motion on the Earth that take gravity into account. 1. Definitions: a. h(t) is the height b. v 0 is the initial vertical velocity (at time zero) c. h 0 is the initial height. (at time zero). Formulas: a. In terms of feet an secons: h(t) = -16t + v 0 t + h 0 b. In terms of meters an secons: h(t) = -4.9t + v 0 t + h 0 K. The erivative is the rate of change function for all types of quantities. L. Marginal means erivative in the worl of economics. We must be careful with units. M. Interpret rectilinear motion when given the graph of the velocity function. N. It is possible to fin where an object is speeing up or slowing own by looking at the graph of y = v(t). O. Assignment: P.130 {8, 9, 10, 1, 0, 3, an 5} 3.5. DERIVATIVES OF TRIGONOMETRIC FUNCTIONS A. Everything is base on two special limits: lim sin = 1 0 B. From the above we can prove: 1. [sin ] = cos lim 0 1 cos = 0

3 . [cos ] = - sin C. The Quotient Rule allows us to prove the remaining trig erivatives. 1. [tan ] = sec. [cot ] = - csc 3. [sec ] = sec tan 4. [csc ] = - csc cot D. Doing everything we have one in the past with trigonometric functions. 1. Tangent an normal lines.. The prouct rule an the quotient rule. 3. Piece-wise ifferentiability an continuity. 4. Higher orer erivatives with occasional patterns. E. Assignment: P. 140 { - 1 even, 8, 9, an 30} 3.6. CHAIN RULE 1 A) Consier the erivatives of y = an y = [g()] g ( ) B) The chain rule for the erivative of composite function y = f(g()) is y = f (g())*g () C) In terms of another notation, if u = g(), then y = f(u), the erivative w.r.t. is: y y u = u D) The part that gets chaine usually eists within parenthesis. You may want to re-write using parenthesis. An eample y = 5 can be re-written as y = ( - 5) (1/) E) For neste parenthesis you have a multiple chains. An eample: f() = sin 4 ( +1) F) Assignment: P. 146 { 31 o an skip } G) An introuction to parametric curves an equations in all of their glory. H) The chain rule allows us to establish the erivative of parametric equations: y y = t t I) Fining the equation of the line tangent to the curve at a point efine by parametric equations. J)The calculator can hanle erivatives of parametric equations. K) Working with a chart like problem #57 is another way to test your unerstaning of the chain rule.

4 L) Assignment: P. 146 {45-48, an 56} 3.7. IMPLICIT DIFFERENTIATION A) An implicit function is one that is not solve for y in terms of. B) To perform implicit ifferentiation w.r.t., just take the erivatives of all variables using the theorems from the past, just stick a (whatever)/ on the en of the non variables. The chain rule allows us to o this. C) Now we can fin the slopes of non-functions such as conic sections. D) At times you must factor out an solve for y. E) Fining horizontal an vertical tangents of implicit functions. F) Look out for proucts an quotients, they are the ownfall of many a stuent. G) When taking the secon erivative of an implicit function you will have to plug in the first erivative. H) We can use implicit ifferentiation to prove the chain rule for rational eponents. I) Unfortunately, the TI-89 oes not o implicit ifferentiation. There eists a program/ownloa for the calculator that enables it to o so. J) Assignment P.155 { - 18 even, 19, 0, 3, 6, 7, 30, 41, an 4} 3.8. DERIVATIVES OF INVERSE TRIGONOMETRIC FUNCTIONS A) Recall how to compute the trig(trig inverse). B) We use implicit ifferentiation to prove the si inverse trig functions: [arcsin ] = 1 1 [arccos ] = 1 1 [arctan ] = [arccot ] = [arcsec ] = 1 1 [arccsc ] = 1 1 C) Note that three are istinctly ifferent while the other three are their opposites. D) Note that y = arcsec an y = arccsc always have a positive slope. This is the reason for the absolute value. We can generate the absolute value of the erivative of arccsc() by consiering the erivative of arcsin(1/). E) Applications involve rates of change of angles with respect to time. Units are in raians/time. These will be consiere in the net chapter. F) Note the interesting relationship between the slope of a function an the slope of its inverse: If (a, f(a)) is a point on y = f () at = a, the erivative of the inverse of f at = a is the reciprocal of the erivative of the function evaluate at = f (a). See a picture an the formula on the bottom of page 57. G) Assignment: P. 16 { - 19 o, skip ; 0, 3}

5 3.9. DERIVATIVES OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS A) Use your calculator to guess what the erivative of y = e is. B) Most of the time we use it chaine up: [eu ] = e u u C) For other bases we can convert to e, which gives us the formula: [au ] = a u (ln a) u D) Recall the formulas for F 1. Compoun Interest: A = P 1+ HG ri K J nt n. Continuous Compoun Interest: A = Pe rt 3. Eponential growth or ecay: A = Ae kt or A = A(b) t/c M 4. Logistic growth: P = 1+ Ae kt E) We can now investigate how fast your money is growing or your car is epreciating. F) We can also consier the rate of growth or ecay in nature. We will o all of this officially in chapter si. G) With our frien implicit ifferentiation we can fin the erivative of the inverse of e [ln u] = 1 u u H) Why stop with the natural logarithm when we can prove: [log au] = 1 u (ln au ) I) We now have virtually all of the erivative formulas we will use in this course! J) Playing log games (epaning as much as possible) will make taking the erivatives of logs much easier. K) Logarithmic ifferentiation (not in the A.P. curriculum) introuces a log to make a ifferentiation possible. L) Assignment: P. 170 { - 43 o an skip, 44, an 47} Manatory Review: P. 17 { - 14 even, 0, 1, 7, 35, 38, 40, 47, 5, 67, 71, 73} 1997 BC Multiple Choice: {, 4, 5, 6, 7, 10, 78, 79}