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Calculus Notes 3.9 & 3.10: Related Rates & Linear Approximations & Differentials.

Calculus Notes 3.9 & 3.10: Related Rates & Linear Approximations & Differentials. Start up:

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Calculus Notes 3.9 & 3.10: Related Rates & Linear Approximations & Differentials.

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  1. Calculus Notes 3.9 & 3.10: Related Rates & Linear Approximations & Differentials. • Start up: • If one side of a rectangle, a, is increasing at a rate of 3 inches per minute while the other side, b, is decreasing at a rate of 3 inches per minute, which of the following must be true about the area A of the triangle? • A is always increasing • A is always decreasing • A is decreasing only when a<b • A is decreasing only when a>b • A is constant • What is the difference between the function L(x) defined in the text and the equation of the tangent line y=f(a)+f’(a)(x-a)? • Answer: • D Answer: 2. There is no difference between L(x) and y = f(a) + f ‘ (x) ( x — a ). Since L(x) is the linear approximation of f at a which uses L(x)= f(a) + f ‘ (x) ( x — a ) to make that approximation.

  2. Calculus Notes 3.9 & 3.10: Related Rates & Linear Approximations & Differentials. • Start up: • If one side of a rectangle, a, is increasing at a rate of 3 inches per minute while the other side, b, is decreasing at a rate of 3 inches per minute, which of the following must be true about the area A of the triangle? • A is always increasing • A is always decreasing • A is decreasing only when a<b • A is decreasing only when a>b • A is constant Example 1: Let’s look at why D for #1. What do we know about triangles and area? a b We have some rates given: Rate of a changing: 3 in/min and Rate of b changing: -3 in/min. Where do we get these from? Take the derivative of the area with respect to time. Since time is not a variable each derivative needs to indicate that. Hence the **/dt in each. Plug in what we know and simplify. So when is dA/dt decreasing? When a>b of course.

  3. Calculus Notes 3.9: Related Rates. • Strategy: • Read the problem carefully. • Draw a diagram if possible. • Introduce notation. Assign symbols to all quantities that are functions of time. • Express the given information and the required rate in terms of derivatives. • Write an equation that relates the various quantities of the problem. If necessary, use the geometry of the situation to eliminate one of the variables by substitution. • Use the Chain Rule to differentiate both sides of the equation with respect to t. • Substitute the given information into the resulting equation and solve for the unknown rate.

  4. Calculus Notes 3.9: Related Rates. Want when t=4h. Example 2: At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35 km/h and ship B is sailing north at 25 km/h. At what rate is the distance between the ships changing at 4:00 PM? 3. Introduce notation. Assign symbols to all quantities that are functions of time. z • Read the problem carefully. y 2. Draw a diagram if possible. 4. Express the given information and the required rate in terms of derivatives. x 150-x 5. Write an equation that relates the various quantities of the problem. If necessary, use the geometry of the situation to eliminate one of the variables by substitution.

  5. Calculus Notes 3.9: Related Rates. Want when t=4h. Example 2: At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35 km/h and ship B is sailing north at 25 km/h. At what rate is the distance between the ships changing at 4:00 PM? z y 6.Use the Chain Rule to differentiate both sides of the equation with respect to t. x 150-x 7. Substitute the given information into the resulting equation and solve for the unknown rate.

  6. Calculus Notes 3.10: Linear Approximations and Differentials. Example 3: Atmospheric pressure P decreases as altitude h increases. At a temperature of 15C, the pressure is 101.3 kilopascals (kPa) at sea level, 87.1 kPa at h=1 km, and 74.9 kPa at h=2 km. Use a linear approximation to estimate the atmospheric pressure at an altitude of 3 km. Use L(x) equation for the linear approximation. In this case it will be: Figure out P ‘ (2): Now plug everything in and solve:

  7. 24cm Calculus Notes 3.10: Linear Approximations and Differentials. • Example 4: The radius of a circular disk is given as 24 cm with a maximum error in measurement of 0.2 cm. • Use differentials to estimate the maximum error in the calculated area of the disk. • What is the relative error? What is the percentage error? • Read the problem carefully. • Draw a diagram if possible. • Introduce notation. Assign symbols to all quantities that are functions of time. • Express the given information and the required rate in terms of derivatives. • Write an equation that relates the various quantities of the problem. If necessary, use the geometry of the situation to eliminate one of the variables by substitution. 6.Use the Chain Rule to differentiate both sides of the equation with respect to t. 7. Substitute the given information into the resulting equation and solve for the unknown rate. So maximum possible error in the calculated area of the disk is about 30cm2.

  8. Calculus Notes 3.10: Linear Approximations and Differentials. • Example 4: The radius of a circular disk is given as 24 cm with a maximum error in measurement of 0.2 cm. • Use differentials to estimate the maximum error in the calculated area of the disk. • What is the relative error? What is the percentage error? (b) Relative error, which is computed by dividing the error by the total (in this case Area). Percentage error, turn the relative error from decimal to percent. PS 3.9 pg.202 #2, 7, 8, 11, 12, 19, 27, 31, 32 (9) PS 3.10 pg.210 #3, 4, 5, 7, 15, 16, 17, 19, 20, 21, 22, 41, 44, 47 (14) Review pg. 214 #1-90 every 3rd problem (30)

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