When does the function assume this value?
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1 Calculus Write your questions and thoughts here! 9.3 Average Value Name: Recall: Average Rate of Change: Mean Value Theorem (MVT) for Derivatives: Notes Average Value of a Function: 1 1. Find the average value of 6 on 1, 3. When does the function assume this value? y f x MVT for Integrals: 1 Comparing average rate of change (secant slope) and average value of a function. Set up the equation for each question, but do not solve it. What units will the ANSWER be? is height (feet) and is time (seconds). a. What is the average height during the first 3 seconds? b. What is the average velocity during the first 3 seconds? 3. 2sin1, where is the rate at which Mr. Brust s waistline is changing (inches per month) and is time (months). a. What is the average rate that Mr. Brust s waistline changes from the 10 th to the 12 th month? b. What is the average change of this rate during the first 5 months?
2 9.3 Average Value of a Function Calculus Name: Find the average value of each function on the given interval. 1. on 2, on 0, 3 3. sin on 0, Practice 4. on 0, on 4, on 3, 1 On the given interval, find the value where the function is equivalent to the average value on that interval on 1, 4 8. on 0, on 5, 2 Find the average rate of change on the given interval on 1, on 1, ln on 1,
3 For 13 14, find where the instantaneous rate of change is equivalent to the average rate of change on 0, on 2, The temperature (in F) hours after 9 AM is approximated by the function sin. Find the average temperature during the time period 9 AM to 9 PM. 16. The depth of water in Mr. Brust s hot tub can be represented by the formula cos 2, where is the time in minutes since he begins pouring in water and is measured in feet. What is the average depth of the water during the first three minutes? Set up the expression and use a calculator to help solve. 17. Find the number(s) such that the average value of 27 on the interval 0, is equal to Find the number(s) such that the average value of 263 on the interval 0, is equal 3. Hint: quadratic formula needed! A Q1 c d Traffic flow is defined as the rate at which cars pass through an intersection, measured in cars per minute. The traffic flow at a particular intersection is modeled by the function defined by 824sin for 030, where is measured in cars per minute and is measured in minutes. (c) What is the average value of the traffic flow over the time interval 1015? Indicate units of measure. (d) What is the average rate of change of the traffic flow over the time interval 1015? Indicate units of measure.
4 A Q3 b d Distance (cm) Temperature ( C) A metal wire of length 8 centimeters (cm) is heated at one end. The table above gives selected values of the temperature, in degrees Celsius ( C), of the wire cm from the heated end. The function is decreasing and twice differentiable. (b) Write an integral expression in terms of for the average temperature of the wire. Estimate the average temperature of the wire using a trapezoidal sum with the four subintervals indicated by the data in the table. Indicate units of measure. (c) Find, and indicate units of measure. Explain the meaning of in terms of the temperature of the wire. (d) Are the data in the table consistent with the assertion that 0 for every in the interval 08? Explain your answer A Q5 d A car is traveling on a straight road. For 0 24 seconds, the car s velocity, in meters per second, is modeled by the piecewise linear function defined by the graph to the right. (d) Find the average rate of change of over the interval Does the Mean Value Theorem guarantee a value of, for 8 20, such that is equal to this average rate of change? Why or why not? 9.3 Average Value of a Function Test Prep 1. The average value of over the interval is (A) 3 3 (B) (C) (D) (E)
5 2. The average value of the function ln on the interval 2, 4 is (A) (B) (C) (D) (E) The function is continuous on the closed interval 1, 3 and has the values given in the table. The equation must have at least two intersections with in the interval 1, 3 if (A) (B) (C) 2 (D) (E) 3 4. A particle moves along the -axis so that its position at time is given by 712. For what value of is the velocity of the particle zero? (A) 2.5 (B) 3 (C) 3.5 (D) 4 (E) The function is given by. On which of the following intervals is increasing? (A), 0 (B), (C) 0, (D) 0, (E),
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