Related Rates and Theorems

This video tutorial covers related rates, providing guidelines and examples to solve problems involving rates of change. It includes examples on calculating the rates of change for revenue, cost, and profit, the area of ripples in a pond, and the speed of an airplane using the Pythagorean theorem. The tutorial emphasizes the use of differentiation and the chain rule.

What is the first step in solving a related rates problem?

• Substitute known values into the equation.

• Solve the equation for the desired rate.

• Identify all given quantities and quantities to be determined.

• Differentiate the equation with respect to time.

In the example of revenue, cost, and profit, what is the rate of change of revenue when x is 10 and dx/dt is 5?

• $200 per day

• $250 per day

• $150 per day

• $50 per day

How is the profit function determined in the revenue, cost, and profit example?

• By multiplying revenue by cost

• By subtracting cost from revenue

• By adding revenue and cost

• By dividing revenue by cost

What is the rate of change of the area of the ripples when the radius is 4 and dr/dt is 0.5 feet per second?

• 12.57 square inches per second

• 12.75 square inches per second

• 10.5 square inches per second

• 15 square inches per second

In the pebble example, what formula is used to find the area of the circle?

• Area = 2πr

• Area = πr^2

• Area = πd

• Area = 2πd

What is the speed of the plane when the distance from the radar is decreasing at 400 mph and s is 10 miles?

• 400 mph

• 700 mph

• 500 mph

• 600 mph

Which theorem is used to relate the distances in the airplane and radar station example?

• Mean value theorem

• Fundamental theorem of calculus

• Binomial theorem

• Pythagorean theorem

In the airplane example, what is the altitude of the plane?

• 6 miles

• 4 miles

• 5 miles

• 7 miles

What is the value of x when s is 10 miles in the airplane example?

• 6 miles

• 8 miles

• 12 miles

• 10 miles

What is the significance of the negative sign in the velocity of the plane?

• It signifies the direction of the plane's velocity.

• It represents the plane's speed is decreasing.

• It shows the plane is moving away from the radar.

• It indicates the plane is slowing down.