



Average Velocity Problems Calculus
In fact, studying these quantities played a major role in the invention of Calculus. Solution manual for calculus 3rd edition by william l. A very useful application of calculus is displacement, velocity and acceleration. As seen in the Angular Velocity of particle section, angular velocity depends on the point that we are measuring the rotation about. Math The height (in feet) of a small weight oscillating at the end of a spring is h(t) = 0. Instead of differentiating velocity to find acceleration, integrate acceleration to find velocity. 1 The Tangent and Velocity Problems. In fact, the slope of the black line gives us the average velocity between 0 and 2 seconds, but that is not what we want. Distance and displacement are both 140 m [East]. Its linear velocity is the cross product of its angular velocity about and its distance from. This calculus video tutorial explains the concepts behind position, velocity, acceleration, distance, and displacement, It shows you how to calculate the velocity function using derivatives and. 2 definitions instantaneous rate of change the instantaneous rate of change of f with respect to x at a is the derivative motion along a line u2013position. 2 s(t) = t2 t 2 4 2 4 6 8 10 12 14 16 Let t be time in seconds and s(t) be distance traveled in feet. Integrate acceleration to get velocity as a function of time. 75 (e) Using your work above and this picture of the graph of the function y(t) given below, answer these true/false questions : (Type in the word "True" or "False"). 75 and b = 2? Include units in b − a a your answer. What is the velocity for all integral times t when acceleration is 0 D. Speed is ignorant of direction. 2 by taking the average of t=0 and t=0. Velocity is the rate of change of displacement (s) so we have: ds/dt = t(2+ + 3t  2) Now usually we would integrate this to find an expression for s in terms of t and use some given initial conditions to find the arbitrary constant of integration but in this problem we are not given any initial condition. pdf), Text File (. We see that the velocity (equivalent to the slope of the tangent of the curve) is not constant. Average Velocity on Brilliant, the largest community of math and science problem solvers. We however, like to think of this as a special case of the rate of change problem. The limit of the average velocity is the derivative, if this limit exists: df lim . is 2 m/s d. Average Velocity  Calculus. (This is calculus. Each problem will walk you through every step so you know how to obtain a solution. Consider an object travelling in a semicircular path from point O to point P in 5 second, as is shown in the Figure. 5] and indicate the corresponding secant line on. AP Calculus AB Review: “Differentiation” Topics on Chapter 2 Test (lessons # 2. 0 ≤ t ≤ 7. It is also worth mentioning that the speed corresponding to a speed is known as the size of the velocity. When is the particle moving to the left? To the right?. This is really the essence of the MVT. 0 \leq t \leq 7. ? A vehicle travels along a line so that its velocity at time t is v(t) = 4t − 8 meters per second. Secants and Tangents. ” Notice that section 1. 2 + 3t^2]i + 5. Set the position, velocity, or acceleration and let the simulation move the man for you. The average velocity over the interval to > s:=t>4. We explain calculus and give you hundreds of practice problems, all with complete, worked out, stepbystep solutions, ALL FREE. And part says compute the average of velocities on the interval t between 0 and 2 and t between 2 and 4. So that's his average velocity, 5 kilometers per hour. Plug t = 2 and t = 3 into the position equation to calculate the height of the object at the boundaries of the indicated interval to generate two ordered pair: (2, 1478) and (3, 1398). The instantaneous velocity is simply the average velocity at a specific instant in time. 1 The Tangent and Velocity Problems. Draw a nice long line with a plastic ruler and compute its slope. Now the idea of average velocity is something that is fairly straightforward, but the idea of instantaneous velocity is a little trickier. If you're seeing this message, it means we're having trouble. Where, x is the distance, which keeps changing with time. Find the average velocity of the object over the interval of time [2, 3]. A cheetah is capable of speeds up to 31 m/s (70 mph) for brief periods. A relative velocity problem can be one of the most difficult problems in a physics course. How to calculate average velocity and speed in onedimension in word problems If you're seeing this message, it means we're having trouble loading external resources on our website. The SI unit is meters per second. These resources generally cover the entire subject of Calculus, and their appropriate sections will be linked in this curriculum. • A derivation and solution using Newton's laws and calculus. Many calculus books will treat this as its own problem. 13 Position, Displacement, and Average Velocity; Instantaneous Velocity and Speed; Acceleration Problem 2 The position as a function of time for a car shooting for a land speed record is. Velocity Problems On this page I put together a collection of velocity problems to help you understand velocity better. We have now seen how to calculate the average velocity between two positions. This lesson does not use integration to solve any of the problems. a) Determine the position and velocity functions for the coin. The focus of your writing should be on clear descriptions and justifications of your methods. Students are expected to explain problems using proper vocabulary and terms. Recall (from Derivative as an Instantaneous Rate of Change) that we can find an expression for velocity by differentiating the expression for displacement:. Problems practice. Remember that velocity is the derivative of position, and acceleration is the derivative of velocity. Find the average velocity of the particle over the interval 0≤t≤5. A summary of Velocity and Acceleration in 's Calculus BC: Applications of the Derivative. Graph the following data on the grid below and answer the questions at the bottom of the page. 1]x [2])/ (2. Definite Integral of a function’s velocity gives the total change in position. In Problems 922, find the average value of/(x). Report an Error. S(55) = 14x(55) 2 +140x(55). Problem 36. Functions (all 6 of them!!!). AP® CALCULUS AB/CALCULUS BC 2015 SCORING GUIDELINES find Bob's average velocity during the interval 0 10. SHOW WORK! Time (sec) Distance (m) 0 50 75 90 110 125 What is the average speed after two seconds? After three seconds? After 5 seconds? What is the average speed between two and four minutes?. The vertical velocity with which the ball hits the ground. First, the change in the ball's momentum by the external force F(t) must be calculated. Suppose the velocity changes several times. 001 (blank for answer. The following practice questions ask you to find the position, velocity, speed, and acceleration of a platypus in …. ly/motioninstraightlinehindi. average velocity of the particle over this interval. Find the minimum acceleration of the particle 2. The change in time is often given as the length of a time interval, and this length goes to zero. Make customizable worksheets about constant (or average) speed, time, and distance, in PDF or html formats. Detailed Solutions, theory and solved examples are provided for each concept. We have now seen how to calculate the average velocity between two positions. Average velocity refers to the ratio of the change in distance and the change in time. A particle moves along the xaxis so that iss velocity at time t, 0≤t≤5, is given by v(t)=3(t1)(t3). If is the position of some moving object, and is time, this section uses the following conventions: is its position function = ′ is its velocity function. We look at average rates of change which become instantaneous, as time intervals become vanishingly small, leading to the notion of a derivative. The average velocity is defined as the change in position divided by the time of travel. However, this technically only gives the object's average velocity over its path. Classify the conic section 25x^2 + 50x + 169y^2 − 676y = 3524. c) Find the instantaneous velocity when t=1 and t=2. Solutions to Calculus Problems  Free download as PDF File (. The position function of a ball thrown into the air with a velocity of 23 ft/sec and initial height of 1 foot is given by the function s(t) 16t2 +23t +1 where s(t) is the height of the ball above the ground after t seconds. a) Find the time subintervals in which the object moves to the right, and those in which it moves to the left. Average velocity is the distance traveled in the time interval divided by the time interval. Velocity is a function of time and defined by both a magnitude and a direction. By using this website, you agree to our Cookie Policy. I think your problem is that your two calculations for the average velocity are different and will generally return different values. At time t=2, the position of the particle is x(2)=0 1. The definition of instantaneous velocity. We explain calculus and give you hundreds of practice problems, all with complete, worked out, stepbystep solutions, ALL FREE. Note that this is different from the average velocity. Average velocity and average speed will fall under the Motion Along A Straight Line topic/ chapter in your textbook. The acceleration of the ball. meters due south in 5. But for irregular shapes the process of integration is used. In calculus, given the position of a body as a function of the time, the instantaneous velocity is the derivative of the position with respect to the time. )(a) Find the average speed over the given time intervals. My textbook offers me the following explanation:. This is really the essence of the MVT. quiz worksheet speed velocity acceleration study com. 7 of the textbook is titled “ TANGENTS, VELOCITIES, AND OTHER RATES OF CHANGE. AP Calculus CHAPTER 5 WORKSHEET INTEGRALS Name Seat # Date More Motion Problems 1. Hence we begin by studying limits and their properties. Calculating Average Speed. Our problem is to determine the velocity of the object after 3 seconds. The direction of average velocity is in the direction of the displacement vector (Δ). Velocity is speed plus direction. Many calculus books will treat this as its own problem. Average value of a function Position, velocity, acceleration problems (calculus) Video transcript  [Voiceover] Let's say that we have a particle that's traveling in one dimension, and its position as a function of time is given as t to the third power plus two over tsquared. Suppose the velocity changes several times. It is a vector quantity. a)Find the velocity and acceleration functions for the stone and draw sign diagrams for each function. what is the average velocity of the car in meters per second for. Find the speed at a given time. The average velocity for the time period between t1 and t2 ( t2 > t1 ), is given by s(t2) − s(t1) t2 − t1 where s(t) is the displacement at time t. We can, however, estimate this instantaneous velocity by computing the average velocity over smaller and smaller intervals containing. On the other hand, a rock dropped straight down into a well has no horizontal velocity, only vertical velocity. The formula of average velocity in calculus becomes more complicated, because the differentiation between the two points is denoted in a different manner. velocity with the formula, it was 48. Make customizable worksheets about constant (or average) speed, time, and distance, in PDF or html formats. average velocity = = i + j Δt x Now, as we usually do in calculus, we let Δt →0. academic assignment help; Affordable Assignment Help; Assignment Help Abu Dhabi; Assignment help Auckland Online Services. 5] time inte EXAMPLE 1 comp avg vel SKILL 10t  1. Calculus Early Transcendentals by Briggs, Cochran, Gillette (We will use the 3rd edition, but any edition will work. The two fundamental problems of calculus will be defined. The graph on the left shows a rectangle whose area is clearly less …. The simplest and most common type of acceleration you will encounter is average acceleration. Also like velocity, acceleration is a vector quantity. Average velocity is a Vector quantity. Distance is 140 m [East] and displacement is 140 m. Average Velocity is calculated as the total distance covered divided by the total time elapsed while travelling in a given direction. Reassessing the problem Let us begin with the above example about the movement of the object P. I believe part d is supposed to be the bracket {1, 1+h} in the limit h tends to zeroThis is the instantaneous velocity or speed. a) Find the time subintervals in which the object moves to the right, and those in which it moves to the left. Below, I will discuss how to calculate acceleration for two cases: average and instantaneous accelerations. Math video on how to estimate instantaneous velocity when position and time are given in table by taking the change in position over change in time using the smallest time interval given in the table of data. The “change in time” gets closer and closer to zero. quiz worksheet speed velocity acceleration study com. Calculus I Homework: The Tangent and Velocity Problems Page 5 Example If a ball is thrown into the air with a velocity of 40 ft/s, its height in feet after t seconds is given by y = 40t−16t 2. A collection of interactive apps to illustrate important concepts taught in AP Calculus AB. Then we'll go through a bunch of questions and just look at the wording to figure out if you're supposed to solve them using the average velocity equation (V avg =ΔX/Δt) or the more common instantaneous velocity equations from kinematics. Draw your best approximation to the secant line to the altitude graph from t =2to t =2. In Problems 922, find the average value of/(x). I have a problem that I am stuck on, The problem states: If a ball is thrown straight up into the air with an initial velocity of 90 ft/s, its height in feet after t seconds is given by y = 90t  16t^2. s = t 3 − 15t 2 + 54t. 51 Thus we see that the average velocity over this interval is 8. Practice calculating average velocity and speed from word problems when an object moves in one direction. Average Velocity Calculation Average Velocity is defined as the displacement over total time. is 2 m/s c. a)Find the velocity and acceleration functions for the stone and draw sign diagrams for each function. Velocity is defined as the rate of change of position with respect to time, which may also be referred to as the instantaneous velocity to emphasize the distinction from the average velocity. Here is a set of practice problems to accompany the Average Function Value section of the Applications of Integrals chapter of the notes for Paul Dawkins Calculus I course at Lamar University. Velocity refers to the instantaneous change in velocity. For 0≤ ≤5, the velocity of the particle is given by 𝑣( )= −2+( 2+3 ) 6 ⁄5− 3, and the position of the particle is given by s(t). Average Velocity Calculation Average Velocity is defined as the displacement over total time. The limit of the average velocity is the derivative, if this limit exists: df lim . But in the case of a different sort of quantity this interpretation does not obviously apply, while the approximation approach always does. briggs Full download link：https://bit. If they are averaged, then the average of the square of a trig function is found to be 1/2, so long as you are taking an integer number of quarter. The bug's velocity, in units per second, is given by the graph below. [2014 Problem 4] Train runs back and forth on an eastwest section of railroad track. Solutions to Calculus Problems  Free download as PDF File (. 5, 1], the displacement is: h = h(1) h(:5) = (58 1 :83 1) (58 :5 :83 (:5)2) = 28:37 The change in time is :5 seconds, so the average velocity is: v ave = 28:37:5 = 56:74 meters per second. 1 The Tangent and Velocity Problems In this section we see how limits arise in trying to ﬁnd the tangent to a curve or (as a special case of that) the (instantaneous) velocity of a. a) The average velocity over the interval [2 sec, 2. Several APtype problems are provided. "Where Δt is the small interval of time. Calculus: Early Transcendentals 8th Edition answers to Chapter 2  Section 2. The average velocity formula describes the relationship between the length of your route and the time it takes to travel. Then we'll go through a bunch of questions and just look at the wording to figure out if you're supposed to solve them using the average velocity equation (V avg =ΔX/Δt) or the more common instantaneous velocity equations from kinematics. 1  The Tangent and Velocity Problems  2. One way to estimate the instantaneous velocity of the car at t =2seconds is to take the average of the slopes of the secant lines before and after t =2. A web page designer creates an animation in which a dot on a computer screen has a position of r = [4. A particle moves along a straight line. a) Determine the position and velocity functions for the coin. Find the marginal average profit function. The height h(t) of the ball after t seconds is given by h(t) = 30t 16t^2 What is the average velocity for the time period beginning when t=1 AND LASTING 0. (If the motion is horizontal, going right is a positive velocity and going left is a negative velocity. Now, can we define. In our graph the car is slowing down but what is its velocity? It is very difficult to evaluate the velocity of the car (because it is continuously changing) but you can try to evaluate an estimate of the velocity, an average velocity. AP Calculus Rectilinear Motion Problems Calculator 1. (a) Find the average velocity for. how do I find the average velocity and instantaneous velocity? This is a question from my Calculus book. (b) Find the value of c such that. 2 Problem 4E. 1 The Tangent and Velocity Problems In this section we see how limits arise in trying to ﬁnd the tangent to a curve or (as a special case of that) the (instantaneous) velocity of a. (a)Suppose that the velocity function of a particle moving a long a coordinate line is v(t) = 3t3 + 2. Velocity, in one dimension, is the measure of the rate of change position per unit time, and is measured in meters per second. Calculus  average velocity. Average velocity is given by: v = Δ s Δ t = s final − s initial t final − t initial. Math video on how to compute the average velocity of an object moving in one dimension from a graph of its position vs. A kangaroo moves in a straigt line so that its velocity at time t 0 on a horizontal line is t t What is the average value of y x2. Velocity Problem. The average velocity of the keychain during this trip can be found using the formula averagevelocity = distancetraveled timetotravel. M 171 Calculus I Spring Semester 2020 This syllabus contains information about this course. A building block for this application is the concept of average velocity. This is the Exam of Calculus of a Single Variable One which includes Increasing and Concave, Piecewise Function, Unrelated Problems, Tangent Line, Domain and Range etc. We look at average rates of change which become instantaneous, as time intervals become vanishingly small, leading to the notion of a derivative. In this foldable, students compare the formulas for velocity, acceleration, speed, average velocity, average speed, and total distance travelled for both types of problems. 5 sec and (ii) is asking for the average velocity between 2 sec and 2. average rate of change (or velocity) is equal to the change in distance divided by the change in time. The instantaneous velocity is represented by the first derivative of the positional equation. The Physical Concept of the Derivative This approach was used by Newton in the development of his Classical Mechanics. IB Mathematics HL. Exercises 1–6 give the positions s = f(t) of a body moving on a coordinate line, with s in meters and t in seconds. Af dt At0 At This is the neat notation that Leibniz invented: Af/At approaches df /dt. Find the average velocity for the time period begining when t = 1 and lasting: a) 0. Calculus TEST: 6. asked by B <3 on September 21, 2008; Physics. Applications of Integration 9. Calculus  average velocity *VELOCITY OF A CAR* Suppose the distance s (in feet) covered by a car moving along a straight road after t sec is given by the function s = f(t) = 2t^2 + 48t. The external force F(t) is a continuous function of time. This question is based upon differential calculus. Average velocity is determined similarly by the commuter and the police officer, determining the change in distance divided by the change in time over which the travel took place. Instructions on solving average velocity based on the average of instantaneous velocities. Velocity is a function of time and defined by both a magnitude and a direction. 5 An object is shot upwards from ground level with an initial velocity of 2 meters per second; it is subject only to the force of gravity (no air resistance). c) Find the instantaneous velocity when t=1 and t=2. MATH 1300: Calculus I 2. The position equation of a particle moving along the xaxis is given by x(t) = t2 – 3t + 2 for t 0. Learn exactly what happened in this chapter, scene, or section of Calculus BC: Applications of the Derivative and what it means. where t denotes the number of seconds since the ball has been thrown and v 0 is the initial speed of the ball (also in meters per second). So for a solid object, the angular velocity of all the particles, from which it is composed, are different. 49 m/sNow I was wrong before when I said that it didn't involve that formula. 1) (3) s s ft/sec, or about 8. In this physics review worksheet, students compare average and instantaneous speed, review Newton's laws, electricity, and magnets. The change in time is 1 second, so the average velocity is: v ave = 57:16:5 = 57:16 meters per second (v) For the time interval [. When the velocity is positive, an article is moving to the right, when it’s negative, it’s moving to the left, and when it’s 0, it’s standing still. ) (a) Find the average velocity during the time period [1, 2]. Instead of differentiating velocity to find acceleration, integrate acceleration to find velocity. Consider an object travelling in a semicircular path from point O to point P in 5 second, as is shown in the Figure. AP Calculus 2. If you run around a circular 400meter track in 80 seconds, and end up back at your starting point, your. What students should be able to do: Approximate the derivative using a difference quotient. In this situation, average velocity is the number of miles traveled divided by the time elapsed, which of course is given in miles per hour. Distinguish between displacement and distance traveled. time by computing slopes of secant lines over shorter and shorter intervals of time. Velocity is referred to as a vector quantity because it has both magnitude and direction. A kangaroo moves in a straigt line so that its velocity at time t 0 on a horizontal line is t t What is the average value of y x2. And suppose you want to nd a line that touches the graph at a certain point so that the slope of the line is the same as the slope of the graph at that point. The instantaneous velocity of the object is the derivative of the position function. This is the Exam of Calculus of a Single Variable One which includes Increasing and Concave, Piecewise Function, Unrelated Problems, Tangent Line, Domain and Range etc. If someone just said "5 kilometers per hour," they're giving you a speed, or rate, or a scalar quantity. compute the stones average velocity over the time interval [0. academic assignment help; Affordable Assignment Help; Assignment Help Abu Dhabi; Assignment help Auckland Online Services. The indefinite integral is commonly applied in problems involving distance, velocity, and acceleration, each of which is a function of time.  eduanswer. Calculate the average velocity and average speed. This is the Exam of Calculus of a Single Variable One which includes Values, Local Minimum, Useful Information, Differentiable, Unrelated Problems, Subintervals etc. 47, the velocity is positive and the acceleration is negative, so the yoyo is slowing down while moving upward (till its velocity. Calculus: Early Transcendentals 8th Edition answers to Chapter 2  Section 2. 6 Use algebra to evaluate the limits in Problems 25— 26. This scenario will give us a very rough approximation of the displacement, but it is a good starting point. As seen in the Angular Velocity of particle section, angular velocity depends on the point that we are measuring the rotation about. 1 seconds (iii) 0. AP® Calculus AB 2015 FreeResponse Questions Number of problems—2 Based on the model B from part (c), find Bob’s average velocity during the interval. (Round your answers to two decimal places. Learn exactly what happened in this chapter, scene, or section of Calculus BC: Applications of the Derivative and what it means. Calculus I Homework: The Tangent and Velocity Problems Page 5 Example If a ball is thrown into the air with a velocity of 40 ft/s, its height in feet after t seconds is given by y = 40t−16t 2. Calculus: Early Transcendentals 8th Edition answers to Chapter 2  Section 2. Find the average velocity of the particle over the interval 0≤t≤5. 1  The Tangent and Velocity Problems  2. 2 definitions instantaneous rate of change the instantaneous rate of change of f with respect to x at a is the derivative motion along a line u2013position. Calculate the girl's average speed for the entire bicycle trip. How is the average velocity of a moving object connected to the values of its position function? How do we interpret the average velocity of an object geometrically on the graph of its position function? How is the notion of instantaneous velocity connected to average velocity? Calculus can be viewed broadly as the study of change. If you are told distance and asked for average velocity, or vice versa, the two are directly related through just the definition of average velocity. We'll also talk about how average rates lead to instantaneous rates and derivatives. Key important points are: Average Value, Unrelated Problems, Evaluate, Even Function, Exact Area, Riemann Sum, Thousands, Measured In Number, Identical Adjacent, Total Area. AP® CALCULUS AB/CALCULUS BC 2015 SCORING GUIDELINES find Bob's average velocity during the interval 0 10. The average velocity of the keychain during this trip can be found using the formula averagevelocity = distancetraveled timetotravel. this expression is also the expression for the slope of a secant line connecting the two points thus we conclude that the average velocity of an object. find the average velocity of the dot between t=0 and t = 2 Give your answer a Help on Physics problem (average velocity)  Science Mathematics. the maximum velocity of the cough occurs when V' = 0. In Problems 18, sketch the curve, find the average value of the function, and sketch the rectangle which has the same area as the region under the curve. MAC2311, Calculus I 2. 1 Exercises  Page 82 7 including work step by step written by community members like you. Hence we begin by studying limits and their properties. Calculate the average velocity and average speed. And part says compute the average of velocities on the interval t between 0 and 2 and t between 2 and 4. 5s? How high is it when t= 2. The Physical Concept of the Derivative This approach was used by Newton in the development of his Classical Mechanics. The total profit (in tens of dollars) from selling x selfhelp books is. The vertical velocity with which the ball hits the ground. Calculating Average Speed. You will need to get assistance from your school if you are having problems entering the answers into your online assignment. 5] and indicate the corresponding secant line on. One example is the path of an airplane. average rate of change of the function over a given interval. It's about 100 km/h much of the time, but sometimes it's around 90 km/h,. O Scribd é o maior site social de leitura e publicação do mundo. Another way to interpret the Mean Value Theorem is to think in terms of slope. However, I also saw a few problems on calculating average velocity over a time interval in projectile motion, and they are calculating it by differentiating (ucosθt i + usinθt0. 81 m/s (or or 32 ft/s) for every second it experiences free fall. Initial value differential equation problems: given the velocity or acceleration with initial condition(s) find the position or velocity. 5gt 2 j), which is the position vector at an instant. You take two points A and B and put a line through them, a secant. A particle moves along the xaxis so that iss velocity at time t, 0≤t≤5, is given by v(t)=3(t1)(t3). The Average Velocity Calculator an online tool which shows Average Velocity for the given input. At the beginning, the slope is `0` (the curve is horizontal, or momentarily flat). What is average speed, what is average velocity, methods to find average speed and average velocity. In some applications the "average velocity" of an object might be needed, that is to say, the constant velocity that would provide the same resultant. (Round your answers to two decimal places. Then the car turns around and travels 40. Now what would you do without a speedometer on your car?. Calculus Section 2. Using calculus, it's possible to calculate an object's velocity at any moment along its path. average velocity calculus problem? If a ball is thrown in the air with a velocity 44 ft/s, its height in feet t seconds later is given by y = 44t  16t2. My textbook offers me the following explanation:. Estimate the distance, d traveled during this period. average velocity  average velocity  average velocity formula  average velocity calculator  average velocity equation  average velocity definition  average. In Problems 2328, find a point c in the given interval such that f(c) is equal to the average value of f(x). Since the interval length is 2, the constant in front of the integral is. Describe the tangent problem and how it led to the idea of a derivative. 6 Tangents and Velocity Problems Tangent Line Suppose you have a graph of a function. The figure shows the velocity of a moving object. If the problem seems to require calculus, explain your reasoning and use a graphical or. Average and instantaneous rate of change of a function In the last section, we calculated the average velocity for a position function s(t), which describes the position of an object ( traveling in. Please read this carefully and keep it for future reference (in case you loose it, a copy of the syllabus will be posted on the class Moodle page). So for a solid object, the angular velocity of all the particles, from which it is composed, are different. On problems 1 and 2, (a) Find the average value of f on the given interval.