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4.6: Related Rates

4.6: Related Rates. Remember this problem?. 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 area will increase. A square with sides x has an area.

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4.6: Related Rates

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  1. 4.6: Related Rates

  2. Remember this problem? 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 area will increase.

  3. A square with sides x has an area What if we were to observe the square getting bigger and each side were increasing at a rate of 0.1 cm/sec cm/sec This would no longer simply be called dx because it is a rate of change. So what should we call it? Hint: It’s a change in length with respect to time. We did this before when talking about units of velocity…

  4. A square with sides x has an area What if we were to observe the square getting bigger and each side were increasing at a rate of 0.1 cm/sec cm/sec At what rate would the area of the square be changing? How we label this will always be decided by… Before we work on that, what would the units for this rate of change be? UNITS! cm2/sec

  5. cm2/sec Given what we’ve just seen, how would we label this change? How would we find ? Try taking the derivative of A with respect to… t

  6. Try taking the derivative of A with respect to… t Remember the Chain Rule… But how? Because this is an instantaneous rate of change... Now let’s find the answer we were trying to find. …this is an exact answer, not an approximation.

  7. Suppose that the radius of a sphere is changing at an instantaneous rate of 0.1 cm/sec. How fast is its volume changing when the radius has reached 10 cm? First, we need the volume formula: Next, what are we looking for? (Remember: How fast is its volume changing?) r= 10 cm Then, what do we know? and… Now what do we do?

  8. The sphere is growing at a rate of . Suppose that the radius of a sphere is changing at an instantaneous rate of 0.1 cm/sec. How fast is its volume changing when the radius has reached 10 cm? r= 10 cm

  9. Find 1 liter = 1000 cm3 Water is draining from a cylindrical tank of radius 20 cm at 3 liters/second. How fast is the water level dropping? “How fast is the water level dropping” means that we need to solve for…

  10. 1 liter = 1000 cm3 Water is draining from a cylindrical tank of radius 20 cm at 3 liters/second. How fast is the water level dropping?

  11. Steps for Related Rates Problems: 1. Draw a picture (sketch). 2. Write down known information. 3. Write down what you are looking for. 4. Write an equation to relate the variables. 5. Differentiate both sides with respect to t. 6. Evaluate.

  12. Truck Problem: Truck A travels east at 40 mi/hr. Truck B travels north at 30 mi/hr How fast is the distance between the trucks changing 6 minutes later? B 6 minutes = A miles miles Since x, y, and z are always changing, they are all variables. So the equation we will have to begin with will be…

  13. Truck Problem: Truck A travels east at 40 mi/hr. Truck B travels north at 30 mi/hr. How fast is the distance between the trucks changing 6 minutes later? B ( ) A

  14. 8 cm A cone filter of diameter 8 cm and height 9 cm is draining at a rate of 2 cm3/min. Find the rate at which the fluid depth h decreases when h = 5 cm. 9 cm What do we know besides the dimensions of the cone? We have three variables That’s one variable too many But Wait! Whatever shall we do?

  15. 8 cm A cone filter of diameter 8 cm and height 9 cm is draining at a rate of 2 cm3/min. Find the rate at which the fluid depth h decreases when h = 5 cm. What do we know besides the dimensions of the cone? That’s one variable too many As we’ve done in the past, let’s see if one variable can be substituted for the other. But Wait!

  16. 8 cm A cone filter of diameter 8 cm and height 9 cm is draining at a rate of 2 cm3/min. Find the rate at which the fluid depth h decreases when h = 5 cm. 4 cm r 9 cm h Now we have two variables! Similar triangles

  17. 8 cm A cone filter of diameter 8 cm and height 9 cm is draining at a rate of 2 cm3/min. Find the rate at which the fluid depth h decreases when h = 5 cm. Will this rate increase or decrease as h gets lower?

  18. 8 cm Show that this is true by comparing both when h = 5 cm and when h = 3 cm. As h gets smaller, gets faster because h is in the denominator.

  19. 8 cm Show that this is true by comparing both when h = 5 cm and when h = 3 cm. Problems like this surface often so remember your geometric relationships such as similar triangles, etc.

  20. Hot Air Balloon Problem: Given: How fast is the balloon rising? Find

  21. Hot Air Balloon Problem: Given: How fast is the balloon rising? Find p

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