m = Average Rate of Change (Secant Slope) Example:

Transcription

1 Average Rate o Change Secant Slope Deinition: The average change secant slope o a nction over a particlar interval [a, b] or [a, ]. Eample: What is the average rate o change o the nction over the interval [0, ]? 0 0 m m Δ Δ or m a a The average rate secant slope over the interval [0, ] is. Derivative o 0

2 Instantaneos Rate o Change Tangent Slope Deinition: The change tangent slope at a particlar point. Here, the interval change in or Δ approaches 0. secant slope tangent slope h m lim h 0 h Eample: What is the instantaneos rate o change o the nction at? h h h h m lim h 0 h h h lim h 0 h h h lim lim h h 0 h h 0 8 The instantaneos rate tangent slope at is 8. Derivative o 0

3 Deinition o the Derivative The derivative o a nction is the instantaneos rate o change at the vale. Graphically, it is tangent slope o the nction. Deinition h lim h 0 h y y prime or prime y y doble prime y y th derivative o y dy d d d Alternative Deinition a lim a a y triple prime dy d or the derivative o y with respect to d d o at or the derivative o at Derivative o 0

4 Dierentiability For a nction to be dierentiable: it mst be continos and its derivative mst be continos. Check the domain o the original nction and its derivative. Occrrences where the Derivative Fails to Eist Discontinity Vertical Tangent: derivative rom each side both approaches or Corner: derivative rom each side dier Csps: derivative rom one side approaches and the other side Derivative o 0

5 Derivative o 0 Power Rle n n n Chain Rle g h g g h Take the derivative rom Inside Ot. Eamples g g g cos sin 0 cos sin 0 cos 6 6 h h

6 Derivative 6 o 0 Prodct Rle v v v Qotient Rle g g g g Eamples tan 6 sec 6 tan sec tan h h sec tan sec tan sec sec sec tan sec sec sec

7 Implicit Dierentiation Steps. Take the derivative derivative or y is y or dy and derivative or is or d.. Isolate all terms with y or dy to one side and everything else to the other. dy. Factor y or dy and solve or y or. d Eample y method y y y y y yy y y yy y y y y y y y y dy method d y y y d dy yd ydy dy d dy ydy dy d d yd dy y d y dy y d y Derivative 7 o 0

8 Derivative 8 o 0 Sqare Root 6 Inverse Fnction I has an inverse and b a, then a b. I is dierentiable at a and c a, then c b. Eample Find the slope o the inverse nction o at. '

9 Derivative 9 o 0 Trigonometric cos sin sin cos sec tan csc cot tan sec sec cot csc csc

10 Derivative 0 o 0 Inverse Trigonometric sin cos tan cot sec csc

11 Derivative o 0 Logarithmic & Eponential e e ln a a a ln a a ln log

12 Derivative o 0 Piecewise nction Take the derivative o each piece separately. Note: The less than or greater than symbols shold not have an eqals sign ater taking the derivative. Eamples < > >, 0, ln,, ln > < < < > <, csc. 0,., 8, cot.,., g g

13 Derivative o 0 Absolte Vale Rewrite as piecewise nction. Solve or inside the absolte vale symbol. The inside o the absolte vale shold be positive with > and negative with <. I qadratic, the inside o the absolte vale shold be negative with the nmbers in between. Then, ollow the steps or a piecewise nction to take the derivative. Eamples < > < <,, 0,,, 6, 6 6 < < < > < < < <.,. or,.,. or,., 6. or, , 0 0 6

14 Etremas Local Relative or Absolte Maimm and Minimms Critical Point or Etremas Critical Points or Points o Inlection End Points 0 0 DNE DNE Derivative o 0

15 st Derivative Line Test Use the irst derivative line test to determine etremas maimms and minimms and intervals o increasing and decreasing vales slope. Steps. Take st derivative, ind critical points, and plot them on a nmber line.. Use vales between to ind intervals the nction is increasing or decreasing.. Sketch the graph, determine etremes and ind the y vales. Eample: 8 End Points: Abs. Ma is at Abs. Min is at Local Ma is 0 at 8 Local Min is 0 at 8 Increasing on the interval, Decreasing on the intervals 8,, 8 DNE : 8 > 0 8 < < 8 0 : 8 0, Derivative o 0

16 nd Derivative Line Test Use the second derivative line test to determine intervals o concavity concave p or down and point o inlections change in concavity. Steps. Take nd derivative, ind critical point, and plot them on a nmber line.. Use vales between to ind intervals the nction is concave p or down.. Determine points o inlection and ind the y vales. Eample: e e e DNE e e : nd derivative is continos 0: e 0 Point o Inlection is, 0.7 Concave Up on the interval, Concave Down on the interval, 0.7 Derivative 6 o 0

17 nd Derivative Test or Etremas I the nction is continos and dierentiable, se the second derivative test or etremas to ind maimms and minimms. Cases or Etremas I c 0 and c < 0, then has a local maimm at c. I c 0 and c > 0, then has a local minimm at c. Eample 6 0: 0, 6 6 Local Ma is at Local Min is at Derivative 7 o 0

18 Interpreting the graph o : intercepts are the Zeros o the Fnction. : Slope increases or decreases. Hills or valleys are Ma or Min. : Crvatre is Concavity. Change in crvatre is Point o the Inlection. Eample : Zeroes are, 0, Local Ma is 0.6 at 0. 6 Local Min is. at. Increasing on the interval, 0.6., Decreasing on the interval 0.6,. Point o Inlection is at 0. Concave Up on the interval 0., Concave Down on the interval, 0. Derivative 8 o 0

19 Interpreting the graph o : Positive and negative y-vales are Slopes. -intercepts are Ma or Min. : Slope o graph is Concavity. Hills or valleys are Points o the Inlection. Note: Use the st and nd Derivative Line Test to nderstand the original nction. Eample Local Ma at Local Ma at Increasing on the interval,, Decreasing on the interval, Point o Inlection at, 0., Concave Up on the interval, 0., Concave Down on the interval, 0., Derivative 9 o 0

20 Interpreting a Table The best way to interpret a table o st and nd derivatives is to perorm the st and nd Derivative Line Tests. This will help to make conclsion abot the original nction. Eample is a continos nction with the ollowing properties. Determine the behavior. < < < < < 6 6 > 6 7 negative 0 positive positive 0 negative positive positive 0 negative negative Local Ma is 7 at 6 Local Min is at Increasing on the interval, 6 Decreasing on the interval, 6, Point o inlection is, Concave p on the interval, Concave Down on the interval, Derivative 0 o 0

21 Odd Fnctions Properties Symmetry abot the origin st derivative will be same nd derivative will be opposite 0, 0 is a point on the graph Even Fnctions Properties Symmetry abot the y-ais st derivative will be opposite nd derivative will be same Derivative o 0

22 Sketching the Original Fnction rom st & nd Derivative Test Steps. Plot any points given.. Sketch with a dashed line sing the st derivative line test.. Following the dashed line, se a solid line showing the crvatre o the graph based on the nd derivative line test. Eample Derivative o 0

23 Mean Vale Theorem I a nction is continos at every point o the close interval [ a, b] and dierential at every point o its interior, the open interval a, b, then there is at least one point c in a, b whose tangent slope is eqal to the average slope rom [ a, b]. tangent slope secant slope Eample b a c Find the mean vale on the interval b a [ 0, ] o. b a c b a c c 0 c 0 c 6 c 7 The mean vale on the interval [ 0, ] is 7 at. Derivative o 0

24 Linearization I is dierentiable at a, then the eqation o the tangent line, L m y, where m a and y a deines the linearization o at a. The approimation L is the standard linear approimation o at a. The point a is the center o the approimation. Eample Find the approimate vale o cos.7 π withot a calclator sing as the center o the approimation. cos L m y π π 0 L 0 sin π L π π L The vale o cos.7 is abot 0.8. Derivative o 0

25 Dierentials Dierentials are sed to estimate approimate change o a nction calclated based on slope where d is the change in the variable. Ths, the dierential dy or d is: Eample dy d d d Inlating a spherical balloon changes its radis rom 6 inches to 6. inches. Use dierentials to estimate the change in the volme. Compare it with the tre change. V r π r V 6 88π V π ΔV 0.6π 88π.6π The tre change in volme is.6π cm. dv π r dr dv π 6 0..π The estimated change in volme is.π cm. The dierence or error is 0.π cm. Derivative o 0

26 Particle in Motion Position: t or s t or y t Velocity: v t t or s t or y t Acceleration: a t v t t or s t or y t Jerk: j t a t v t t or s t or y t harmonic motion only Eample A particle moves along the -ais. Its position is given by t t sin t cos t. For what vales o t, 0 < t <, is the particle moving right? Find the total distance traveled by the particle over the interval 0,. t t sin t cos t v t t cos t sin t sin t t cos t a t t sin t cos t 0 t cos t t 0,.7,.7 A particle moves to the right when it s derivative is positive, which is rom 0,.7.7,. The total distance the particle traveled is Derivative 6 o 0

27 Modeling and Optimization Steps. Sketch a diagram.. Write down all cles, qestions, ormlas.. Solve by inding minimm or maimm.. Check yor answer. Does it make sense? Eample Yo are designing a rectanglar poster to contain 0 in o printing with a -in. margin at the top and bottom and a -in. margin at each side. What overall dimensions will minimize the amont o paper? A WL 0 WL 0 L W 0 L 0 A W L 8 A WL 8W L 0 0 A W 8W W W 00 A 8W 8 W 00 A 8 W W W ± W 9 L in by 8 in Derivative 7 o 0

28 Cost & Proit Derivates can be sed to ind the etra revene r, cost c, or proit p reslting rom selling or prodcing one more item. This is called marginal revene r, marginal cost c, or marginal proit p. r the revene rom selling items c the cost o prodcing items p r c the proit rom selling items A company breaks even when the proit p eqals zero. Eample Sppose r 8 represents revene and c represents cost, where measred in thosands o nits. Is there a prodction level that maimizes proit? I so, what is it? p r c 8 p nits Derivative 8 o 0

29 Related Rates Are eqations involving two or more variables that are dierentiable nction o time t. To solve, write an eqation to show the relationship between the two variables. Dierentiate the eqation with respect to time. Eample Water rns into a conical tank at the rate o 9t /min. The tank stands point down and has a height o 0t and a base radis o t. How ast is the water level rising when the water is 6t deep? r V π r h r h h 0 dv dt h π h π h dh π h dt The water level rises abot 0. t/min as the tank ills at a rate o 9t /min. dh 9 π 6 dt dh 9 9π dt dh 0. dt π Derivative 9 o 0

30 Derivative 0 o 0 L Hôpital s Rle I by sing direct sbstittion the limit eqals an indeterminate orm or 0 0 lim g a, then take the limit as a o the derivate o the top and bottom ntil the limit eists. lim lim g g a a Eamples 0 lim 0 0 lim lim lim lim lim ln lim