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Section 2.6: Related Rates

Section 2.6: Related Rates. Introduction to Related Rates. We have seen a lot of relations (whether implicit or explicit) that involve two variables (frequently x and y ). It is possible these two variables are themselves functions of another variable, such as t . For instance:.

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Section 2.6: Related Rates

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  1. Section 2.6: Related Rates

  2. Introduction to Related Rates We have seen a lot of relations (whether implicit or explicit) that involve two variables (frequently x and y). It is possible these two variables are themselves functions of another variable, such as t. For instance:

  3. Introduction to Related Rates As x and y change, their rates of change are related to each other. But how are they related? Notice how when t changes, both the x and y change in relation to the value of t. Let’s investigate what occurs when t changes:

  4. Introduction to Related Rates Differentiate both sides Chain Rule Twice Now we know how the rate of change for x and y are related to each other. In our exercises, we will not need to know the exact relations. In order to take the derivative of the relation using x and y , it must be done with the respect to t. For instance:

  5. Example 1 Find the derivative by differentiating both sides. Chain Rule Substitute the known information Solve for the unknown Suppose x and y are both differentiable functions of t and are related by . Find when x = 10, if when x = 10.

  6. Example 2 Find the derivative by differentiating both sides. Find other important values: x Chain Rule Substitute the known information Solve for the unknown Suppose x and y are both differentiable functions of t and are related by . Find when x = 9, if when x = 9 and y>0.

  7. Example 3 Find the derivative by differentiating both sides. Chain Rule Substitute the known information ft3 per minute Solve for the unknown A spherical balloon is being filled with a gas in such a way that when the radius is 2ft, the radius is increasing at the rate 1/6 ft/min. How fast is the volume ( ) changing at this time?

  8. Related Rates Guidelines • Draw a figure, if appropriate, and assign variables to the quantities that vary. (Be careful not to label a quantity with a number unless it never changes in the problem) • Find a formula or equation that relates the variables. (Eliminate unnecessary variables) • Differentiate the equations. (typically implicitly) • Substitute specific numerical values and solve algebraically for any required rate. (The only unknown value should be the one that needs to be solved for.)

  9. Example 1 Find the rates by differentiating both sides. 20 ft 6 ft Chain Rule y x Substitute the known information Using similar triangles, the equation is: Solve for the unknown A person 6 ft tall is walking away from a streetlight 20 ft high at the rate of 7 ft/s. At what rate is the length of the person’s shadow increasing? ft/s

  10. Example 2 Find the rates by differentiating both sides. 5 m y Chain Rule Ladder Substitute the known information x Find other important values: Using The Pythagorean Theorem, the equation is: Solve for the unknown A bag is tied to the top of a 5 m ladder resting against a vertical wall. Suppose the ladder begins sliding down the wall in such a way that the foot of the ladder is moving away from the wall. How fast is the bag descending at the instant the foot of the ladder is 4 m from the wall and the foot is moving away at the rate of 2 m/s? m/s x

  11. Example 3 Find the rates by differentiating both sides. 10 ft 8 ft Nothing is known about b… b 3 ft h Using similar triangles: Chain Rule Substitute the known information Using the volume of a prism, the equation is: Solve for the unknown ft/min A trough 10 ft long has a cross section that is an isosceles triangle 3 ft deep and 8 ft across. If water flows in at the rate 2 ft3/min, how fast is the surface rising when the water is 2 ft deep?

  12. Example 4 Find the rates by differentiating both sides. This is “x” and there is no “x” in the derivative… x Chain Rule Substitute the known information Θ 25 mi Use “x” to find other important values: Using The Trigonometry, the equation is: Solve for the unknown A rocket launches with a velocity of 550 miles per hour. 25 miles away there is a photographer filming the launch. At what rate is the angle of elevation of the camera changing when the rocket achieves an altitude of 25 miles? rad/h

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