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2019 Related Rates

2019 Related Rates. AB Calculus. Known Limits:. GOAL : to find the rates of change of two (or more) variables with respect to a third variable (the parameter) This is a adaptation of IMPLICIT functions x and y are implicit functions of t . ILLUSTRATION :

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2019 Related Rates

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  1. 2019 Related Rates AB Calculus

  2. Known Limits:

  3. GOAL: to find the rates of change of two (or more) variables with respect to a third variable (the parameter) This is a adaptation of IMPLICIT functions x and yare implicit functions oft . ILLUSTRATION: A point is moving along the parabola, Find the rate of change of ywhen x = 1 if xis changing at2 units per second. Intro: time moving moving graph dy

  4. 1). DRAW A PICTURE! – Determine what rates are being compared.  2). Assign variables to all given and unknownquantities and rates.  3). Write an equation involving the variables whose rates are given or are to be found ·Equation of a graph? ·Formula from Geometry?   The equation must involve only the variables from step 2. – ((You may have to solve a secondary equation to eliminate a variable.))  4). Use Implicit Differentiation (with respect to the parameter t).  5). AFTER DIFFERENTIATION, substitute in all known values (( You may have to solve a secondary equation to find the value of a variable.)) PROCEDURE: May plug in a constant as long as it is unchanging

  5. Geometry formulas: Sphere: Cylinder: Cone: Pythagorean Theorem:

  6. METHOD: Inflating a Balloon - 1 A spherical balloon is inflated so that the radius is changing at a rate of 3 cm/sec. How fast is the volume changing when the radius is 5 cm.? Draw and label a picture. Step 1: List the rates and variables. Find an equation that relates the variables and rates. (Extra Variables?) When r =5 Plugin 5 gives vol not rate of change Differentiate (with respect to t.) Plug in and solve.

  7. Ex 2: Ladder w/ secondary equation A 25 ft. ladder is leaning against a vertical wall. If the bottom of the ladder is pulled horizontally away from the wall at a rate of 3 ft./sec., how fast is the top of the ladder sliding down the wall when the bottom is 15 ft. from the wall? # constant does not change ever you can plug in the equation The ladder is coming down -2.25 ft/sec The ladder is coming down

  8. II: Similar Triangles

  9. Similar Triangles B D E A F C Similar Triangles may be the whole set up. Similar Triangles may be required to to eliminate an extra variable – or- to find a missing value

  10. Ex 4: A person is pushing a box up a 20 ft. ramp with a 5 ft. incline at a rate of 3 ft.per sec.. How fast is the box rising? derivative 20 ft 5 z y x as

  11. Getting smaller Ex 5: Pat is walking at a rate of 5 ft. per sec. toward a street light whose lamp is 20 ft. above the base of the light. If Pat is 6 ft. tall, determine the rate of change of the length of Pat’s shadow at the moment Pat is 24 ft. from the base of the lamppost. The distance of top of shadow from post How fast is the tip of Pat’s shadow changing 20 6 x y 6 6 y-x y

  12. Ex 6: Cone w/ extra equation Three variables Water is being poured into a conical paper cup at a rate of cubic inches per second. If the cup is 6 in. tall and the top of the cup has a radius of 2 in., how fast is the water level rising when the water is 4 in. deep? Too many variables need to find r r changes h changes Only two variables

  13. III: Angle of Elevation

  14. hyp opp adj

  15. Angles of Elevation SOH – CAH - TOA c a θ b Hint: The problem may not require solving for an angle measure … only a specific trig ratio. ie. need sec θinstead of θ 5 3 θ 4

  16. A balloon rises at a rate of 10 ft/sec from a point on the ground 100 ft from an observer. Find the rate of change of the angle of elevation of the balloon from the observer when the balloon is 100 ft. above the ground. Ex 7: When y =100 100 or 100

  17. A fishing line is being reeled in at a rate of 1ft/sec from a bridge 15ft above the water. At what rate is the angle between the line and the water changing when 25ft of line is out. Ex 8: z When z = 25 ft 15 25 15 20

  18. Ex 9: A television camera at ground level is filming the lift off of a space shuttle that is rising vertically according to the position function , where yis measured in feet and t in seconds. The camera is is2000 ft. from the launch pad. Find the rate of change of the angle of elevation of the camera 10 sec. after lift off. y 2000ft y y 5000 2000

  19. IV: Using multiple rates

  20. Ex 11: If one leg, AB, of a right triangle increases at a rate of 2 in/sec while the other leg, AC, decreases at 3 in/sec, find how fast the hypotenuse is changing when AB is 72 in. and AC is 96 in. B z y x C A

  21. 5: AP Questions

  22. Example 12: AP Type At 8 a.m. a ship is sailing due north at 24 knots(nautical miles per hour) is a point P. At 10 a.m. a second ship sailing due east at 32 knots is a P. At what rate is the distance between the two ships changing at (a) 9 a.m. and (b) 11 a.m.?

  23. A right triangle has height 7 cm and the hypotenuse is increasing at a rate of 2 cm/sec. When the hypotenuse is 25 cm, find: a). the rate of change of the base. b). The rate of change of the acute angle at the base, c). The rate of change of the area of the triangle. Ex 13: AP Type

  24. Last Update • 11/12/11 • BC:

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