Study Guide for Test 4

Below is a study guide to help you focus your preparations for Test 4.  There are are ten questions total in this study guide but only nine of them will be selected to be on the actual test.  Good Luck in your studies.

Question #1        You will need to know the relationships, in terms of both derivatives and anti-derivatives, between the position, velocity, and acceleration functions.  For example, velocity can be thought of as the anti-derivative of acceleration.  Velocity can also be thought of as the derivative of the position function.  This question will be a word problem similar to the ones found in section 4.10.  If you can independently do problems #68 - #71 from section 4.10 you will be adequately prepared for this question on the test.

Question #2        Realize that the area under the velocity function (assuming that it is positive) gives the distance traveled by the particle.  Knowing this is helpful when you have a table of values where the independent variable is time and the dependent variable is velocity.  You can use the information in the table of values to estimate the distance traveled by an object.  Example 4 on page 322 gives an illustration on how this is done using rectangles.  Study this example as well as similar problems from section 5.1 to be successful on this question.  You need to know how to construct rectangles with left-hand endpoints, right-hand endpoints, and mid-points.  For example, you may be required to estimate the distance traveled using 4 rectangles with left-hand endpoints or 3 rectangles with mid-points.  When studying example 4 and similar problems, be sure to experiment with various sample points with different numbers of rectangles.

Question #3        Know how to find anti-derivatives of functions subject to initial conditions.  When computing indefinite integrals (anti-derivatives) remember to add the constant of integration.  Remember that the purpose of initial conditions is to pin down exactly what the constant of integration will be.  Example 7 on page 304 shows how this is done.  Be sure to practice problems #53 - #58 in section 4.10 in preparation for this question on the test.  Be aware that some problems involving initial conditions are more difficult than others and may require more algebraic manipulations in order to solve them.

Question #4        The function  below is called an area function.  It gives the area under the graph of the function f(t) from a fixed point a to the variable x.  For each value of x, there is only one value for g(x); this is why it is a function.   

You need to be able to compute the area function for different values of  x given a picture of the graph of  f(t), the integrand.  This is usually done by looking at the graph and realizing that the area under the "curve" is some familiar geometric object that has an area formula for.  For example, if  f(t) were a linear function then the area under it could be broken up into triangles and rectangles for which there are simple area formulas e.g. the area of a triangle is "one half base times height."  You need to be able to find intervals of increase and decrease for an area function as well as local extreme values.  You should also be able to sketch the graph of an area function given the graph of the integrand.  You may want to refer to problems #2 and #3 on pages 347-348 for extra practice.  In addition, know how to sketch the graph of an anti-derivative for a given function.  For practice on this see problems #39-43 on page 306.  

Question #5        For this question, be prepared to know how to find areas under curves using the limit definition of a definite integral.  You will not be allowed to use the Fundamental Theorem of Calculus for this question.  This definition is given two-thirds of the way down on page 326.  You DO NOT need to memorize the formulas 4, 5, and 6 found on page 329 as these will be given to you on the test.  You should be able to find areas under linear and quadratic functions using this definition.  An example for how this process is done is given on page 329 at the very bottom, called Example 2.  We did similar examples in class using a multi-step procedure with many details that you may want to review.  To make things simple, we usually take the sample point in the definition of definite integral to be the right-hand endpoint.  With this in mind, remember that "delta x" equals "(b - a) /n" and that the "ith endpoint of the ith subinterval or simply 'x sub i' " equals "a + i*delta x."  Also we mentioned in class that with respect to the summation symbol (capitol Greek letter sigma), the variable n is constant but not so with respect to the limit.  This is important to know this when actually computing the limit because expressions involving the variable n can be brought out in front of the sigma symbol.  Additional problems for practice are problems #21 - 25 on page 337.

Question #6        You need to be able to recognize a given limit as a definite integral on a specified interval.  Practice problems that address this are #20 and 21 on page 325.  In these problems you must identify  f(x)  as well as the interval [a, b]. 

Question #7        Know the difference between distance and displacement.  Displacement is the change in position from the starting point to the stopping point.  Its possible to travel a distance of 6 miles yet have a displacement of zero because you can go somewhere for three miles and come back the same way you came stopping at the same place you started from.  You can find displacement of a moving object by integrating the velocity function from the starting time to the stopping time.  The starting and stopping times are your limits of integration in this case.  Finding distance is more tricky because you need to take into account the direction a moving object travels.  An object moving in the positive direction will have positive velocity while an object moving in a negative direction will have negative velocity.  In the real world, velocity functions must be continuous.  This means for an object moving in a straight line to change directions, the object must first come to a complete stop before moving in the opposite direction.  To measure distance, you measure the displacement of an object from the starting time to the first time the velocity of the object is zero.  Then you measure the the displacement from the first time the velocity is zero to the second time the velocity is zero and so on until you come to the stopping time.  Since distance is always non-negative, you must count each displacement as if it were positive even though you may get a negative displacement calculation wise.  You can do this by taking the absolute value of each of the displacements.  Finally you must add up all the displacements after adjusting via taking their absolute values.  This sum will be the total distance traveled by the moving object.  For practice, please refer to #53 - 56 on page 358. 

Question #8        Know how to use the Fundamental Theorem of Calculus to evaluate definite integrals.  You must first find an anti-derivative of the integrand and evaluate it at the upper limit of integration while subtracting the same anti-derivative evaluated at the lower limit of integration. 

Question #9        Be sure you know how to use the substitution rule for evaluating both definite and indefinite integrals.  The substitution rule for integration is analogous to the chain rule for differentiation.  Problems #7 - 32 on page 366 deal with the substitution rule for indefinite integrals while problems #37 - 54 deal with the substitution rule for definite integrals.  Be sure to seek help from either me or the math support center if you are having trouble with the substitution rule.  We worked out problems in class with great many details.  Be sure and study these examples.

Question #10       You need to know how to use the power rule for finding anti-derivatives of power functions.  This rule is given on page 301.  The power rule works not only when n is a natural number but when n is a fraction and negative number as well.  In general, it works when n is ANY real number.  You also need to know how to find derivatives of area functions using the Fundamental Theorem of Calculus Part I as found boxed in RED on page 342.  Examples 3 and 4 on page 343 and 344 show how to to do this.  Additional practice problems are found on page 348 problems #7 - 18.  You should also realize that the variable we integrate with respect to can be anything.  For example, you are used to seeing "dx" in most integrals.  One can change "dx" to "dy" or "d&" or anything else you wish provided you change the same variable in the integrand as well.  This is what is referred to as a "dummy" variable.