Problem of the Day If x 2 + y 2 = 25, what is the value of d 2 y at the point (4,3)? dx 2 a) -25/27 c) 7/27 e) 25/27 b) -7/27 d) 3/4

Problem of the Day If x 2 + y 2 = 25, what is the value of d 2 y at the point (4,3)? dx 2 a) -25/27 c) 7/27 e) 25/27 b) -7/27 d) 3/4 y = (25 - x 2 ) 1/2 d y = ½(25 - x 2 ) -½ (-2x) dx d 2 y = (-x)[-½(25 - x 2 ) -3/2 (-2x)] + (-1)(25 - x 2 ) -½ d x 2 d2 y = -25/27 dx 2

Related Rates Given: y = x X and Y are related by this equation Remember that the derivative is a rate of change Can we differentiate with respect to time?

Given: y = x Related Rates Find dy/dt when dx/dt = 2 and x = 1 dy = 2x dx dt dy = 2(1)(2) = 4 dt substitute known quantities

Related Rates Problem Strategy 1. Draw a picture and name the variables and constants. Use t for time. 2. Write down the numerical information in terms of the variables you have chosen. 3. Write down what you are asked to find using the variables you have chosen. 4. Write an equation that relates the variables. 5. Differentiate with respect to t. 6. Evaluate.

You are blowing a bubble with bubble gum and can blow 3 in 3 /s of air into the bubble. How fast is the radius r increasing with respect to time when the radius is 1 inch?

r v = volume of bubble at time t r = radius of bubble at time t 1. Draw a picture and name the variables and constants.

r v = volume of bubble at time t r = radius of bubble at time t dv/dt = 3 in 3 /s r = 1 2. Write down the numerical information in terms of the variables you have chosen. You are blowing a bubble with bubble gum and can blow 3 in 3 /s of air into the bubble. How fast is the radius r increasing with respect to time when the radius is 1 inch?

r v = volume of bubble at time t r = radius of bubble at time t dv/dt = 3 in 3 /s r = 1 3. Write down what you are asked to find using the variables you have chosen. Find dr/dt

You are blowing a bubble with bubble gum and can blow 3 in 3 /s of air into the bubble. How fast is the radius r increasing with respect to time when the radius is 1 inch? r v = volume of bubble at time t r = radius of bubble at time t dv/dt = 3 in 3 /s r = 1 Find dr/dt 4. Write an equation that relates the variables. V = 4/3 Π r 3

You are blowing a bubble with bubble gum and can blow 3 in 3 /s of air into the bubble. How fast is the radius r increasing with respect to time when the radius is 1 inch? r v = volume of bubble at time t r = radius of bubble at time t dv/dt = 3 in 3 /s r = 1 Find dr/dt V = 4/3 Π r 3 5. Differentiate with respect to t. dv = 4/3 Π (3r 2 dr) dt dv = 4 Π r 2 dr dt dv 1 = dr dt 4Π r 2 dt

You are blowing a bubble with bubble gum and can blow 3 in 3 /s of air into the bubble. How fast is the radius r increasing with respect to time when the radius is 1 inch? r v = volume of bubble at time t r = radius of bubble at time t dv/dt = 3 in 3 /s r = 1 Find dr/dt V = 4/3 Π r 3 dv = 4/3 Π (3r 2 dr) dt dv = 4 Π r 2 dr dt dv 1 = dr dt 4Π r 2 dt 6. Evaluate. (3) 1 = dr 4Π(1) 2 dt 3 = dr 4Π dt

r v = volume of bubble at time t r = radius of bubble at time t dv/dt = 4 in 3 /s r = 3 Find dr/dt V = 4/3 Π r 3 dv = 4/3 Π (3r 2 dr) dt dv = 4 Π r 2 dr dt dv 1 = dr dt 4Π r 2 dt 6. Evaluate. Suppose you increase your effort when r = 3 and blow in 4 in 3 /s of air. How fast is the radius now increasing?

r v = volume of bubble at time t r = radius of bubble at time t dv/dt = 4 in 3 /s r = 3 Find dr/dt V = 4/3 Π r 3 dv = 4/3 Π (3r 2 dr) dt dv = 4 Π r 2 dr dt dv 1 = dr dt 4Π r 2 dt 6. Evaluate. (4) 1 = dr 4Π(3) 2 dt 1 = dr 9Π dt Suppose you increase your effort when r = 3 and blow in 4 in 3 /s of air. How fast is the radius now increasing?

How fast is the volume V increasing with respect to the radius when the radius r is 1 inch? v = volume of bubble at time t r = radius of bubble at time t dv/dt = 3 in 3 /s r = 1 Find dv/dr V = 4/3 Π r 3

How fast is the volume V increasing with respect to the radius when the radius r is 1 inch? v = volume of bubble at time t r = radius of bubble at time t dv/dt = 3 in 3 /s r = 1 Find dv/dr V = 4/3 Π r 3 dv = 4/3 Π (3r 2 dr) dr dv = 4 Π r 2 dr dv = 4 Π (1) 2 d r

Water is drained out of a conical tank. The volume, V, the radius, r and height, h of water are all functions of time. Find the related rate equation.

V = Π r 2 h 3 dv = Π [r 2 dh + h 2r dr ] dt 3 dt dt

A pebble is dropped in a pond, causing ripples of concentric circles. The radius r of the outer ripple increases at a rate of 1 foot per second. When r = 4, what is the rate of change of area of disturbed water?

Given: Find: A = Π r 2 dr = 1 dt dA when r = 4 dt r

A pebble is dropped in a pond, causing ripples of concentric circles. The radius r of the outer ripple increases at a rate of 1 foot per second. When r = 4, what is the rate of change of area of disturbed water? Given: Find:A = Π r 2 dr = 1 dt dA = Π [ 2r dr ] dt A = Π r 2 dA when r = 4 dt dA = Π (2)(4)(1) = 8Π dt

Remember to substitute after differentiating. Look at the table on top of page 151 for other examples of mathematical models.

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