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Rate of Change and Instantaneous Velocity
Section 2.2A Calculus AP/Dual, Revised Β©2017 11/24/ :54 AM Β§2.2A: Rates of Change
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Average Velocity Rates of change play a role whenever we study the relationship between two changing quantities. A familiar example is velocity, which is the rate of change of position with respect to time. If an object is traveling in a straight line, the average velocity over a given time interval from point π¨ to point π© We cannot define instantaneous velocity as a ratio because we would have to divide by the length of the time interval, which is zero. However we can estimate the instantaneous velocity by computing the average velocity over successively smaller intervals. 11/24/ :54 AM Β§2.2A: Rates of Change
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Average Rate of Change The average rate of change describes the constant rate at which a function would have to change over an interval if the function were to achieve the same vertical displacement over the length of the interval investigated. Average Velocity Formula: π = βπ βπ = π π βπ π πβπ = πͺπππππ ππ π·πππππππ π»πππ π¬ππππππ
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Steps in Average Rate of Change
Plug your π and π in to get the corresponding π π and π π . Plug into the formula and reduce. Average Rate of Change uses only endpoints of the interval. It does not reflect any fluctuations within the interval. Label. 11/24/ :54 AM Β§2.2A: Rates of Change
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Example 1 A car travels along a straight road and the driver observes the mile markers along the trip as shown in the table. What are the units for the average speed over the 24 minutes? Explain your answer. Time (in min.) π π ππ ππ Mile Marker ππ ππ ππ ππ 11/24/ :54 AM Β§2.2A: Rates of Change
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Example 2 Water is pumped into a tank at a rate modeled by πΎ π =ππππ π β π π /ππ liters per hour for πβ€πβ€π, where π is measured in hours. Using the table below, estimate πΉ β² π and show all work that leads to the answer. Explain your answer. π (hours) π π π π π πΉ π (Liters/Hr) ππππ ππππ πππ πππ πππ 11/24/ :54 AM Β§2.2A: Rates of Change
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Example 2 Water is pumped into a tank at a rate modeled by πΎ π =ππππ π β π π /ππ liters per hour for πβ€πβ€π, where π is measured in hours. Using the table below, estimate πΉ β² π and show all work that leads to the answer. Explain your answer. π (hours) π π π π π πΉ π (Liters/Hr) ππππ ππππ πππ πππ πππ 11/24/ :54 AM Β§2.2A: Rates of Change
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Your Turn Jo Ann jogs along a straight path. For πβ€πβ€ππ, Joβs speed is given by a differential function π π where π is measured in minutes and π π is measured in meters per minute. Using the table below, estimate π β² ππ and show all work that leads to the answer. Explain your answer and label accordingly. π (Min) π π2 ππ ππ ππ π π (Met/Min) πππ πππ βπππ πππ 11/24/ :54 AM Β§2.2A: Rates of Change
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Example 3 A tennis ball is dropped from a height of 100 feet, its height π at time π is given by the position function, π
π = βπππ π +πππ where π
is measured in feet and π is measured in seconds. Find the average velocity over the time interval of π, π . 11/24/ :54 AM Β§2.2A: Rates of Change
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Example 3 A tennis ball is dropped from a height of 100 feet, its height π at time π is given by the position function, π
π = βπππ π +πππ where π
is measured in feet and π is measured in seconds. Find the average velocity over the time interval of π, π . 11/24/ :54 AM Β§2.2A: Rates of Change
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Example 4 If an object is dropped from a tall building, then the distance it has fallen after π seconds is given by the function π
π = βπππ π . Find its average speed (average rate of change) between π and π+π seconds. 11/24/ :54 AM Β§2.2A: Rates of Change
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Your Turn Suppose you are driving to a friendβs house. After 30 minutes, you are 15 miles from your house. After 3 hours, you are 60 miles from your house. What is your average rate of change between these times (in hours)? Label accordingly. 11/24/ :54 AM Β§2.2A: Rates of Change
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Instantaneous Velocity
If we look at a graph of the position of the object, a secant line can be drawn through the points ππ, π π and ππ, π π . The average velocity of the object would be the slope of the secant line through ππ, π π and ππ, π π . If the secant line from ππ is moved closer to t1, the secant line gets closer and closer to becoming a tangent line to the graph at ππ, π π . The slope of the tangent line is the instantaneous velocity of the object when the time is ππ The instantaneous rate of change at is the limit of the average rates of change as gets closer and closer to We must consider values of on both the left side of and on the right side of as we take the limit. An expression such as βπ βπ = π π βπ π πβπ is called a difference quotient. 11/24/ :54 AM Β§2.2A: Rates of Change
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Instantaneous Velocity
Velocity of an object at time π is found by taking the DERIVATIVE of the position function. VELOCITY is a vector, which means it has direction. So, velocity can be positive, negative, or zero. SPEED is the absolute value of velocity, and cannot be negative. Instantaneous Velocity Formula: π₯π’π¦ βπβπ π π+βπ βπ π βπ =πβ² π =π π . It is also known as π
π π
π . 11/24/ :54 AM Β§2.2A: Rates of Change
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Average Velocity vs. Instantaneous Velocity
Solve using Algebra Slope - (Not much Calculus needed) πβ² π Take the Derivative 11/24/ :54 AM Β§2.2A: Rates of Change
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Example 5 At time π=π, a diver jumps from a platform diving board that is ππ feet above the water. The position of the diver is given by π π = βπππ π +πππ+ππ where s is measured in feet and π is measured in seconds. Graph the function on the graphing calculator and sketch the result below. Find the average velocity (average rate of change) at which the diver is moving for the time interval π, π.π . Use your answers to (b) to estimate the instantaneous rate of change (instantaneous velocity) at which the diver is moving at time π =π second. When does the diver hit the water? What is the diverβs velocity at impact? What is their speed? When does the diver stop moving upward and start their descent? 11/24/ :54 AM Β§2.2A: Rates of Change
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Example 5a At time π=π, a diver jumps from a platform diving board that is ππ feet above the water. The position of the diver is given by π π = βπππ π +πππ+ππ where π is measured in feet and π is measured in seconds. (a) Graph the function on the graphing calculator and sketch the result below. 11/24/ :54 AM Β§2.2A: Rates of Change
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Example 5b At time π=π, a diver jumps from a platform diving board that is ππ feet above the water. The position of the diver is given by π π = βπππ π +πππ+ππ where π is measured in feet and π is measured in seconds. (b) Find the average velocity (average rate of change) at which the diver is moving for the time interval π, π.π . 11/24/ :54 AM Β§2.2A: Rates of Change
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Example 5c At time π=π, a diver jumps from a platform diving board that is ππ feet above the water. The position of the diver is given by π π = βπππ π +πππ+ππ where π is measured in feet and π is measured in seconds. (c) Use your answers to (b) to estimate the instantaneous rate of change (instantaneous velocity) at which the diver is moving at time π=π second. 11/24/ :54 AM Β§2.2A: Rates of Change
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Example 5c At time π=π, a diver jumps from a platform diving board that is ππ feet above the water. The position of the diver is given by π π = βπππ π +πππ+ππ where π is measured in feet and π is measured in seconds. (c) Use your answers to (b) to estimate the instantaneous rate of change (instantaneous velocity) at which the diver is moving at time π = 1 second. 11/24/ :54 AM Β§2.2A: Rates of Change
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Example 5d At time π=π, a diver jumps from a platform diving board that is ππ feet above the water. The position of the diver is given by π π = βπππ π +πππ+ππ where π is measured in feet and π is measured in seconds. (d) When does the diver hit the water? 11/24/ :54 AM Β§2.2A: Rates of Change
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Example 5e At time π=π, a diver jumps from a platform diving board that is ππ feet above the water. The position of the diver is given by π π = βπππ π +πππ+ππ where π is measured in feet and π is measured in seconds. (e) What is the diverβs velocity at impact? What is their speed? 11/24/ :54 AM Β§2.2A: Rates of Change
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Example 5f At time π=π, a diver jumps from a platform diving board that is ππ feet above the water. The position of the diver is given by π π = βπππ π +πππ+ππ where π is measured in feet and π is measured in seconds. (f) When does the diver stop moving upward and start their descent? 11/24/ :54 AM Β§2.2A: Rates of Change
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Example 6 The temperature π», in degrees Fahrenheit, of a cold potato placed in a hot oven is given by π»=π π , where π is the time in minutes since the potato was put in the oven. What is the meaning of the statement π β² ππ =π? 11/24/ :54 AM Β§2.2A: Rates of Change
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AP Free Response Question 1 (non-calculator)
π π π π π ππ π(π) π π βπ π Let π be a function that is twice differentiable for all real numbers. The table above gives values of π for selected points in the closed interval, π, ππ . Estimate π β² π . Show the work that leads to the answer. 11/24/ :54 AM Β§2.2A: Rates of Change
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AP Free Response Question 2 (non-calculator)
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AP Free Response Question 2 (non-calculator)
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Assignment Page 114 93-97 all, 105 11/24/2018 10:54 AM
Β§2.2A: Rates of Change
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