Drill: find the derivative of the following • 2xy + y2 = x + y • 2xy’ +2y + 2yy’ = 1 + y’ • 2xy’ + 2yy’ – y’ = 1 – 2y • y’(2x + 2y – 1) = 1 – 2y • y’ = (1-2y)/(2x + 2y -1) • xsin(y)= 1 – xy • xcos(y)y’ + sin (y) = -(xy’ +y) • xcos(y)y’ + sin (y) = -xy’ –y • xcos(y)y’+xy’= -y – sin (y) • y’ (xcos (y) + x) = -y – sin (y) • y’ = – (y + sin(y))/(xcos(y) + x)

Related Rates Section 4.6

Objectives • Students will be able to • solve related rate problems.

Strategy for Solving Related Rate Problems • Understand your problem • Develop a mathematical model of the problem • Write an equation relating the variable whose rate of change you seek with the variables whose rate of change you know • Differentiate both sides of the equation implicitly with respect to time t. • Substitute values for any quantities that depend on time • Interpret the solution

Finding related rate equations • Assume the radius r of a sphere is a differentiable function of t and let V be the volume of the sphere. Find an equation that related dV/dt and dr/dt

Finding related rate equations • Assume that the radius and height h of a cone are differentiable functions of t and let V = volume of the cone. Find an equation that relates dV/dt, dr/dt and dh/dt.

Example 1 Finding Related Rate Equations Assume that the width a and the length b of a rectangle are functions of t and let A be the area of the rectangle. Find an equation that relates

Example 1 Finding Related Rate Equations Assume that the width a and the length b of a rectangle are functions of t and let A be the area of the rectangle. Find an equation that relates

Example 2 A Rising Balloon A hot-air balloon rising straight up from a level field is tracked by a range finder 700 feet from the lift-off point. At the moment the range finder’s elevation angle is , the angle is increasing at the rate of 0.16 radians per minute. How fast is the balloon rising at that moment?

Example 2 A Rising Balloon A hot-air balloon rising straight up from a level field is tracked by a range finder 700 feet from the lift-off point. At the moment the range finder’s elevation angle is , the angle is increasing at the rate of 0.16 radians per minute. How fast is the balloon rising at that moment?

Example 3 A Highway Chase A police cruiser approaching a right-angled intersection from the north is chasing a speeding car that has turned the corner and is now moving straight east. When the cruiser is 0.70 mile north of the intersection and the car is 0.80 mile to the east, the police determine with radar that the distance between them and the car is increasing at 15 mph. If the cruiser is moving at 45 mph at the instant of measurement, what is the speed of the car?

Example 3 A Highway Chase A police cruiser approaching a right-angled intersection from the north is chasing a speeding car that has turned the corner and is now moving straight east. When the cruiser is 0.70 mile north of the intersection and the car is 0.80 mile to the east, the police determine with radar that the distance between them and the car is increasing at 15 mph. If the cruiser is moving at 45 mph at the instant of measurement, what is the speed of the car?

Example 3 A Highway Chase A police cruiser approaching a right-angled intersection from the north is chasing a speeding car that has turned the corner and is now moving straight east. When the cruiser is 0.70 mile north of the intersection and the car is 0.80 mile to the east, the police determine with radar that the distance between them and the car is increasing at 15 mph. If the cruiser is moving at 45 mph at the instant of measurement, what is the speed of the car? Let x = distance of the speeding car from the intersection Let y = distance of the police cruiser from the intersection Let z = distance between the car and the cruiser

Example 3 A Highway Chase We want to know: We know:

Example 3 A Highway Chasex = .8 and y = .6 We can now substitute, remembering that z2 = x2 + y2 ; so z =(x2+ y2)1/2 At that moment, the car’s speed was about 59.3 mph.

Example 4 Filling a Conical Tank Water runs into a conical tank at the rate of 12 ft3 /min. The tank stands point down and has a height of 12 ft and a base radius of 9 ft. How fast is the water level rising when the water is 8 ft deep?

Example 4 Filling a Conical Tank Water runs into a conical tank at the rate of 12 ft3/min. The tank stands point down and has a height of 12 ft and a base radius of 9 ft. How fast is the water level rising when the water is 8 ft deep? r h

Homework • Page 251: 9-21 odd