RATES OF CHANGE: GEOMETRIC.

Slides:

Advertisements

Similar presentations

1 Related Rates Section Related Rates (Preliminary Notes) If y depends on time t, then its derivative, dy/dt, is called a time rate of change.

Advertisements

4.6 Related Rates.

 A cylinder has two identical flat ends that are circular and one curved side.  Volume is the amount of space inside a shape, measured in cubic units.

4.6 Related Rates What you’ll learn about Related Rate Equations Solution Strategy Simulating Related Motion Essential Questions.

Related Rates Chapter 3.7. Related Rates The Chain Rule can be used to find the rate of change of quantities that are related to each other The important.

3.11 Related Rates Mon Dec 1 Do Now Differentiate implicitly in terms of t 1) 2)

Section 2.8 Related Rates Math 1231: Single-Variable Calculus.

A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute,

Do now: These cuboids are all made from 1 cm cubes

3.11 Related Rates Mon Nov 10 Do Now

1 Related Rates Finding Related Rates ● Problem Solving with Related Rates.

Related Rates M 144 Calculus I V. J. Motto. The Related Rate Idea A "related rates" problem is a problem which involves at least two changing quantities.

By: Kyler Hall Autrey: AP Calculus March 7, 2011.

Section 4.1: Related Rates Practice HW from Stewart Textbook (not to hand in) p. 267 # 1-19 odd, 23, 25, 29.

R ELATED R ATES. The Hoover Dam Oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant.

APPLICATION OF DIFFERENTIATION AND INTEGRATION

Ch 4.6 Related Rates Graphical, Numerical, Algebraic by Finney Demana, Waits, Kennedy.

Warm-Up If x 2 + y 2 = 25, what is the value of d 2 y at the point (4,3)? dx 2 a) -25/27 c) 7/27 e) 25/27 b) -7/27 d) 3/4.

2.6 Related Rates I. Volume changes-implicit With change in volume, volume, radius, and height of the volume in a cone all depend upon the time of the.

Section 4.6 Related Rates.

Related Rates. The chain rule and implicit differentiation can be used to find the rates of change of two or more related variables that are changing.

6.5: Related Rates Objective: To use implicit differentiation to relate the rates in which 2 things are changing, both with respect to time.

Miss Battaglia AP Calculus Related rate problems involve finding the ________ at which some variable changes. rate.

A box with a square base and top has a volume of 10 meters cubed. The material for the base costs $12 per square meter and the material for the sides.

1 §3.4 Related Rates. The student will learn about related rates.

4.1 - Related Rates ex: Air is being pumped into a spherical balloon so that its volume increases at a rate of 100cm 3 /s. How fast is the radius of the.

3.11 Related Rates Tues Nov 10 Do Now Differentiate implicitly in terms of t 1) 2)

4.6: Related Rates. A square with sides x has an area If a 2 X 2 square has it’s sides increase by 0.1, use differentials to approximate how much its.

1 Related Rates Finding Related Rates ● Problem Solving with Related Rates.

DO NOW Approximate 3 √26 by using an appropriate linearization. Show the computation that leads to your conclusion. The radius of a circle increased from.

Mr. Moore is pushing the bottom end of a meter stick horizontally away from the wall at 0.25m/sec. How fast is the upper end of the stick falling down.

Related Rates 3.6.

3.9 Related Rates In this section, we will learn: How to compute the rate of change of one quantity in terms of that of another quantity. DIFFERENTIATION.

Section 2.8 Related Rates. RELATED RATE PROBLEMS Related rate problems deal with quantities that are changing over time and that are related to each other.

Find if y 2 - 3xy + x 2 = 7. Find for 2x 2 + xy + 3y 2 = 0.

2.6: Related Rates Greg Kelly, Hanford High School, Richland, Washington.

Related Rates. We have already seen how the Chain Rule can be used to differentiate a function implicitly. Another important use of the Chain Rule is.

Warm up 1. Calculate the area of a circle with diameter 24 ft. 2. If a right triangle has sides 6 and 9, how long is the hypotenuse? 3. Take the derivative.

MATH 1910 Chapter 2 Section 6 Related Rates.

Related Rates with Area and Volume

Section 2-6 Related Rates

DERIVATIVES WITH RESPECT TO TIME

Sect. 2.6 Related Rates.

Implicit Differentiation Problems and Related Rates

MTH1170 Related Rates.

Calculus I (MAT 145) Dr. Day Monday Oct 16, 2017

Calculus I (MAT 145) Dr. Day Monday Oct 23, 2017

Calculus I (MAT 145) Dr. Day Friday Oct 20, 2017

Calculus I (MAT 145) Dr. Day Friday, October 5, 2018

Implicit Differentiation

Section 2.6 Calculus AP/Dual, Revised ©2017

4.1: Related Rates Greg Kelly, Hanford High School, Richland, Washington.

8-1 Volume of Prisms & Cylinders

4.6: Related Rates Greg Kelly, Hanford High School, Richland, Washington.

Math 180 Packet #9 Related Rates.

Related Rates Chapter 5.5.

AP CALCULUS RELATED RATES

Section 3.5 – Related Rates

Calculus I (MAT 145) Dr. Day Wednesday February 27, 2019

Use Formula An equation that involves several variables is called a formula or literal equation. To solve a literal equation, apply the process.

AP Calculus AB 5.6 Related Rates.

Calculus I (MAT 145) Dr. Day Monday February 25, 2019

Calculus I (MAT 145) Dr. Day Monday March 4, 2019

§3.9 Related rates Main idea:

Lesson 4.6 Core Focus on Geometry Volume of Cylinders.

Lesson 4.8 Core Focus on Geometry Volume of Spheres.

Lesson 4.8 Core Focus on Geometry Volume of Spheres.

Related Rates AP Calculus Keeper 26.

Related Rates ex: Air is being pumped into a spherical balloon so that its volume increases at a rate of 100cm3/s. How fast is the radius of the balloon.

Presentation transcript:

RATES OF CHANGE: GEOMETRIC

Rates of Change Take the derivative with respect to whatever the variable is in the equation. Ex) Find the rate of change of area of a circle with respect to radius when the radius is 1 cm. A: Area r: Radius

Related Rates A word problem that are solved by implicit differentiation, relation is always in respect to time. They have to do with changes in two or more variables with respect to time. The variables are related such that when one changes the other must also change.

Step One Ex) You have a spherical snowball and allow it to melt. Changes occur both to its radius and volume as it melts. We know that the volume is related to the radius by . Since volume and radius change over time, take the derivative of the volume formula with respect to time. Rate of change of volume Rate of change of radius Derivative with respect to time. with respect to time equals

Step Two 2 takes the place of r and 1.5cm\min takes the place of . Now say we want to find the rate of change of volume when the radius is 2cm and the radius is changing at 1.5cm\minute. 2. Identify where the given values should go within the related rates

Step Three 3. Use algebra to solve for missing variable Answer is:

Example One! Find the rate of change of volume, with respect to side length of a cube when the side length is 8cm.

Example 2 Water is flowing out of a cylindrical storage tank at a rate of 2 . If the tank has a radius of 8m, how fast is the water level falling? because the radius doesn’t change

A water tank is in the shape of a cone with a height of 5m and a diameter of 6 m at the top. Water is being pumped into the tank at a rate of 1.6 . Find the rate at which the water level is rising when the water is two meters deep. Example 3

SUMMARY Related rates are word problems that are solved by implicit differentiation. The relation is always in respect to time. The first step in a related rates problem is to take the derivative. Next, identify where the given values should go with the equation. Lastly, remember to use algebra to solve for the missing variable. Keep track of what equations to use. Try not to get area confused with volume. Remember to use correct units in your answer. Hints: draw pictures and label them; write down equations that relate the variables in the problem; DIFFERENTIATE; substitute in the information and solve.