22: Applications of Differential Calculus

Transcription

1 22: Applications of Differential Calculus A: Time Rate of Change The most common use of calculus (the one that motivated our discussions of the previous chapter) are those that involve change in some quantity over time. While dy dx this case (x would represent time), we often will change the variables say, dh dt height over time, or dv dt for change in velocity over time can still be used in for change in B: General Rates of Change Note that if the variables have positive association (they increase and decrease together), then the derivative will be positive. If one variable increases when the other decreases, the derivative will be negative. C: Motion in a Straight Line A common (introductory) application of calculus is motion on a straight line. The original function represents the position (x-coordinate) of a moving object. Typically, this function is given as s( t ). The First Derivative The first derivative then gives the instantaneous velocity positive value indicating motion to the right, and negative values indicating motion to the left. To find average velocity, you must be given an initial an a final position. Note that a velocity of zero indicates that the object is at rest. Speed and velocity are not the same thing! Velocity is magnitude and direction speed is. magnitude only. Thus, the instantaneous speed of a moving object is s ( t) The Second Derivative The second derivative gives instantaneous acceleration positive values indicating increasing velocity, and negative values indicating decreasing velocity. Be careful "increasing velocity" does not necessarily mean "going faster!" Consider the following graph of velocity versus time: Figure 1 - Velocity vs. Time Between times zero and three, the velocity of the object is increasing. Since the velocity is non-zero, the object is not at rest. When t= 3, the velocity is zero, and the object is at rest. How would you describe the motion of an object that is moving, then comes to rest? HOLLOMAN S IB HIGHER LEVEL YEAR 2 HH NOTES 22, PAGE 1 OF 7

2 You'd say it was decelerating! Even though the velocity is increasing! So be careful. When velocity is positive, increasing and decreasing velocity have their usual meanings (accelerate and decelerate); when velocity is negative, the usual meanings are reversed To find average acceleration, you must be given an initial and final velocity. Distance Traveled You can also determine the distance traveled by the object. The difference between the initial position and the final position can be called the net distance traveled but is usually called the displacement. The total distance traveled (or gross distance traveled) is the sum of the distances traveled in each direction. Here consider this graph of position versus time: Figure 2 - Position vs. Time The position at t= 1 is zero; the position at 5 t= is zero. Thus, the displacement for t [ 1,5] is zero. The total distance traveled is different the object first moves about 3 units to the right; then changes direction and moves about 6 units to the left (for a total of 9 units); then turns once more and moves about 3 units to the right, for a total of 12 units. For displacement, we add the movements with the signs intact; for total distance traveled, we add the absolute values of the movements. D: Some Curve Properties Monotonicity A function f ( x ) is monotonic on the interval ( a, b ) if f ( x) does not change sign (from positive to negative, or vice versa) on that interval. For example, if it is rising or flat, but doesn't fall that's monotonic. If the function changes direction (on that interval), then it's not monotonic. Increasing/Decreasing A function f ( x ) is increasing on the interval (, ) x < x, then f ( x ) < f ( x ). a b means that for any pair of x values in the interval, if 1 s 1 2 Another way to say this is that the function is increasing on the interval if the first derivative is positive for all values in the interval. HOLLOMAN S IB HIGHER LEVEL YEAR 2 HH NOTES 22, PAGE 2 OF 7

3 Those who actually bother to read the text will note a discrepancy some of us insist that increasing (and decreasing) are properties of open intervals, while others insist that closed intervals are fine. I belong to the former group. There are similar definitions for intervals where the function is decreasing I'll let you figure them out. A sign diagram (of the first derivative) is a great way to determine intervals where a function is increasing and decreasing. Maxima/Minima If a function is increasing, then switches to decreasing, what happens in-between? If a function is decreasing, then switches to increasing, what happens in-between? The answers actually depend on the function, and whether or not it is a nice function a continuous function. For the moment, let us only consider functions that are continuous on the set of real numbers. In this case, an increasing function (positive derivative) changing to a decreasing function (negative derivative) must pass through some place where it is neither increasing nor decreasing a place where the derivative is zero! Since the function was rising, then started falling, it must have achieved a local maximum (singular; plural is maxima). f x has a local maximum at x c a, b so that every x ( ) = if there exists some open interval ( ) value in that interval causes f ( x) f ( c). In other words, there is some (small) region where f ( c ) is bigger than any other f ( x ). f ( x ) has a global maximum at x c = if f ( x) f ( c) for all values of x in the domain of f. Again, there are similar definitions for local and global minima Note that a maximum or minimum must (for now) be a location where the derivative equals zero; however, just because the derivative equals zero, that does not mean that there is an extreme value (maximum or minimum). Horizontal/Stationary Inflections Sometimes, the function levels off (derivative equals zero), but it doesn't change direction. The text calls this a horizontal (or stationary) inflection. E: Rational Functions A rational function is one that can be written as a ratio of polynomials. Many of the details involved in graphing rational functions involve (or touch on topics from) calculus. Thus, here are some steps you should follow Factor the numerator and denominator. That will make everything else easier. Look for factors in common between the numerator and denominator. Common factors will manifest themselves as holes (or removable discontinuities). These common factors cancel out for most values of x, and thus have (mostly) no effect on the graph. The value of x that makes the factor zero is not in the domain of the function, and thus causes a hole in the graph. Having noted the x-coordinate of the hole, go ahead and cancel out those common factors as I just mentioned, they only have an effect at that one value of x. HOLLOMAN S IB HIGHER LEVEL YEAR 2 HH NOTES 22, PAGE 3 OF 7

4 Any values of x that make the denominator equal to zero are not in the domain, and will manifest themselves as vertical asymptotes. Any values of x that make the numerator equal to zero will cause the function to have a value of zero thus, these are the locations of x-intercepts. If zero is in the domain of the function, then plugging in an x-value of zero will give the location of the y-intercept. That leaves just one item which the text breaks into a bunch of cases, even though there is one guiding principle! Specifically, the final item of consideration is end behavior what happens to the function for values of x that are (very) far from zero? The guiding principle is this divide the rational function (polynomial long or synthetic short take your pick) out into a quotient and remainder. For very large (far from zero) values of x, the rational function will behave/look like the quotient from the division. When dividing the rational function, there are three (or four) possible results, depending on the degrees of the numerator and denominator. If the denominator is of higher degree, then the rational function cannot be divided the quotient is zero, and the remainder is the function itself. In this case, the end behavior is a horizontal asymptote at y = 0. Let me emphasize that a horizontal asymptote is a very different creature from a vertical asymptote (it is unfortunate that they are both called asymptotes). A vertical asymptote is a domain issue there is a value that simply cannot be used; thus, we get a line that cannot be touched, crossed, etc. However, a horizontal asymptote is end behavior it is an indication of the tendency that the function has. In this case, the asymptote can be crossed at will the only requirement is that the function approach that line as x moves away from zero. Back to the possible cases from the division If the numerator and denominator have equal degrees, then the division will result in a constant. In this case, the end behavior is a horizontal asymptote at that constant. If the numerator is of higher degree than the denominator, then the quotient will be a polynomial. In this case, I would say that we have and end behavior asymptote at that polynomial. It is this last case that some break into two cases specifically, some will separately consider the case where the degree of the numerator is exactly one greater than the denominator. That would result in a linear quotient, which is often called an oblique asymptote (or, if you are a bit backwards, a slant asymptote). My opinion: why (at this point) treat the lines any differently from the other polynomials? I said that this relates to calculus and it does! I just didn't make the connections for you F: Inflections and Shape Type Concavity There are two types of curve: concave up, and concave down. HOLLOMAN S IB HIGHER LEVEL YEAR 2 HH NOTES 22, PAGE 4 OF 7

5 Figure 3 - Concavity A curve is concave up at those locations where its second derivative is positive; it is concave down where its second derivative is negative. Points of Inflection A Point of Inflection is a location where the concavity of the curve (and thus, the sign of the second derivative) changes. G: Optimization This is the process of finding and verifying locations where a function has an extreme value. If you've got a graphing calculator, then you could just look at the graph so I will focus my comments on the algebraic methods. Identifying Candidates First of all, we have to determine where the extreme values might occur. Begin by taking the derivative. Any points where the tangent is horizontal (the first derivative equals zero) are candidates. Also (technically), any points where the first derivative is undefined are candidates. This is because of a type of feature called a cusp. Figure 4 - Cusps These should be pretty unusual better to be safe and check for them, even if you never run across one. Also any domain restrictions will create additional candidates. If the function is restricted to a, b, then x= a and x= b are also candidates. a closed interval [ ] In summary: potential locations of extrema are where the first derivative equals zero where the first derivative is undefined the endpoints of the domain (if the domain is a closed interval) Checking Having identified all candidates for extrema, now you've got to check and see which ones are, in fact, extreme values. HOLLOMAN S IB HIGHER LEVEL YEAR 2 HH NOTES 22, PAGE 5 OF 7

6 If the domain is restricted to a closed interval, then just plug the candidates back into the original function, and compare the results the largest is the maximum, and the smallest is the minimum. If you're not working on a closed interval, then you have some options. Sign Chart: examine the sign of the first derivative on either side of the candidate. If the first derivative is positive, then negative, then the candidate must be a maximum. If the derivative changes from negative to positive, then the candidate is the location of a minimum. Second Derivative Test: look at the sign of the second derivative for the candidate value. If the second derivative is positive, then the function is concave up, and the candidate is a minimum. If the second derivative is negative, the function is concave down, and the candidate is a maximum. Note that if the second derivative is zero, then you're back to square one you'll have to do something else to determine if the candidate is the location of an extreme value. Optimization This term is a really fancy way of saying "word problem." This means that you may have to write the function on your own! H: Economic Models Some details about some specific types of calculus problem (of major interest to business majors) Cost A Cost Function y C( x) = gives the cost (y) of producing a certain number (x) of some item. This should be an increasing function it is best when this function is concave down (the rate of change in cost is decreasing). C( x) The Average Cost Function is y=. x The Marginal Cost is the instantaneous rate of change of cost with respect to the number of C x ). items produced (how fast the cost is changing as you increase production; ( ) Demand, Revenue and Profit The Demand Function gives the price (per item) if x items are sold. Typically, a business will charge less per item as the number of items increases thus, the demand function should be a decreasing function. The Revenue Function gives the revenue produced by selling x items. This is the product of x and the demand function. This should be an increasing function! Its derivative is the marginal revenue function. The Profit Function is the difference in the revenue and cost functions. Its derivative is the marginal profit function. Of prime interest is maximum profit which will occur when the marginal profit is zero. HOLLOMAN S IB HIGHER LEVEL YEAR 2 HH NOTES 22, PAGE 6 OF 7

7 I: Implicit Differentiation So far, we have exclusively dealt with explicit functions functions where the y-variable can be separated from x completely; hence, the form y = for our functions. Of course, this is not the only way to do it what about the equation of a circle? This equation is implicit x and y are mixed together. Okay, okay that's not a function. It is a perfectly fine curve, and it makes perfect sense to discuss the slope of the tangent at a point! How do you use the definition in order to take limits and find the slope of the tangent line? The answer is to do everything the same, but treat y as another function (and don't forget the chain rule!). For example, 2 y is really a function raised to a power which, by the general power rule, has dy a derivative of 2y (the dy is from the chain rule). dx dx We'll do some examples of this in class HOLLOMAN S IB HIGHER LEVEL YEAR 2 HH NOTES 22, PAGE 7 OF 7