- Express changing quantities in terms of derivatives.
- Use the chain rule to find the rate of change of one quantity that depends on the rate of change of other quantities.
- Chapter 4.1 Related Rates
- p. 350: 1, 5, 10, 17, 20, 25, 29
- Application-Related Rates
Exit problems of the session
The radius of sphere increases at a rate of 2m/s. Find the rate at which the volume increases when the volume is 36.
A vertical cylinder is leaking water at a rate of 1ft/sec. If the cylinder has a height of 10 ft and a radius of 1 ft, at what rate is the height of the water changing when the height is 6ft?
Sand pours from a chute and forms a conical pile whose height is always equal to its base diameter. The height of the pile increases at a rate of 5 ft/hr. Find the rate of change of the volume of the sand in the conical pile, when the height of the pile is 4 ft.
- A 13-ft ladder is leaning against a wall. If the top of the ladder slides down the wall at a rate of 2ft/sec, how fast is the bottom moving along the ground when the bottom of the ladder is 5 ft from the wall?
- Solving a related-rates problem:
- To solve a related rates problem, first draw a picture that illustrates the relationship between the two or more related quantities that are changing with respect to time. Assign symbols to all variables involved in the problem.
- In terms of the variables, state the information given and the rate to be found.
- Find an equation relating the variables.
- Use differentiation, applying the chain rule as necessary, to find an equation that relates the derivatives.
- Substitute all known values into the equation from step 4, then solve for the unknown rate of change.
- Useful formulas:
circumference of a circle: area of a circle:surface area of a sphere: volume of a sphere:volume of a cylinder:volume of a cone:
Videos and Practice Problems of Selected Topics
- Approaching cars (6:52) As two cars approach the same intersection from different roads, how does the rate of change of the distance between them change?
- Falling ladder (5:48) You’re on a ladder. The bottom of the ladder starts slipping away from the wall. Analyze the rate of change.
- Water pouring into a cone (11:31) As you pour water into a cone, how does the rate of change of the depth of the water relate to the rate of change in volume?
- * Practice: Related rates problems using the Pythagorean Theorem. (4 problems)
- * Practice: Related rates advanced problems. (4 problems)