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Objectives: Be able to find the derivative of an equation with respect to various variables.

Related Rates. Objectives: Be able to find the derivative of an equation with respect to various variables. Be able to solve various rates of change applications. Critical Vocabulary: Derivative, Rate of Change. I. Derivatives.

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Objectives: Be able to find the derivative of an equation with respect to various variables.

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  1. Related Rates • Objectives: • Be able to find the derivative of an equation with respect to various variables. • Be able to solve various rates of change applications. Critical Vocabulary: Derivative, Rate of Change

  2. I. Derivatives Example 1: Find the derivative of y with respect to x: x2 + y2 = 25 Example 2: Find the derivative of x with respect to y: x2 + y2 = 25

  3. I. Derivatives Example 3: Find the derivative of y with respect to t: x2 + y2 = 25 Example 4: Find the derivative of x with respect to t: x2 + y2 = 25

  4. I. Derivatives Example 5: Find dy/dt when x = 2 of the equation 4xy = 12 given that dx/dt = 4 (If x = 2, Then y = 3/2)

  5. Page 304 #1-7 odd

  6. Related Rates • Objectives: • Be able to find the derivative of an equation with respect to various variables. • Be able to solve various rates of change applications. Critical Vocabulary: Derivative, Rate of Change WARM UP: Find dy/dt: 3x2y3 = 12

  7. I. Derivatives Warm Up: Find dy/dt: 3x2y3 = 12

  8. II. Applications Guidelines for Solving Related Rate Problems 1. Identify all given quantities and quantities to be determined. • Make a sketch and label your diagram • Write an equation involving the variables whose rates or change either are given or are to be determined. • Volume Formulas (Inside Cover of Book) • Area Formulas (Inside Cover of Book) • Pythagorean Theorem (when you sketch looks like a RT Δ) 3. Using implicit Differentiation, differentiate with respect to time. 4. Substitute Values as necessary. Then solve for the required rate of change.

  9. II. Applications Example 6: A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius (r) of the outer ripple is increasing at a constant rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area (A) of the disturbed water changing? What do I know: What do I need to find: Differentiate: The total area of the disturbed water is changing at 25.13 ft2/sec. Substitute:

  10. II. Applications Example 7: Air is being pumped into a spherical at a rate of 4.5 cubic feet per minute. Find the rate of change of the radius when the radius is 2 feet. What do I know: The rate of change of the radius when the radius is 2 feet is 0.09 ft/min. What do I need to find: Differentiate: Substitute:

  11. Page 304-306 #13-23 odd

  12. Related Rates • Objectives: • Be able to find the derivative of an equation with respect to various variables. • Be able to solve various rates of change applications. Critical Vocabulary: Derivative, Rate of Change

  13. II. Applications Example 8: A ladder 10 feet length is leaning against a brick wall. The top of the ladder is originally 8.5 feet high. The top of the ladder falls at a fixed rate of speed dy/dt. As time goes by the distance x(t) from the base of the wall to the bottom of the ladder changes. What is the rate of change of the distance x(t)?

  14. II. Applications Example 8: A ladder 10 feet length is leaning against a brick wall. The top of the ladder is originally 8.5 feet high. The top of the ladder falls at a fixed rate of speed dy/dt. As time goes by the distance x(t) from the base of the wall to the bottom of the ladder changes. What is the rate of change of the distance x(t)? What do I know: What do I need to find: Substitute: Differentiate:

  15. II. Applications Example 9: A baseball diamond has the shape of a square with sides 90 feet long. Suppose a player is running from 1st to 2nd at a speed of 28 feet per second. Find the rate at which the distance from home plate is changing when the player is 30 feet from second base.

  16. II. Applications Example 9: A baseball diamond has the shape of a square with sides 90 feet long. Suppose a player is running from 1st to 2nd at a speed of 28 feet per second. Find the rate at which the distance from home plate is changing when the player is 30 feet from second base. What do I know: What do I need to find: Substitute: Differentiate:

  17. Page 304-306 #27, 33 • Worksheet: “Applications: Rate of Change”

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