1_Functions_merged.pdf

Functions Dr. Nobphadon Suksangpanya Mathematics I – Dr.Nobphadon Suksangpanya 1  FUNCTIONS: A function f is a rule that associates a unique output with each...

Functions Dr. Nobphadon Suksangpanya Mathematics I – Dr.Nobphadon Suksangpanya 1  FUNCTIONS: A function f is a rule that associates a unique output with each input. If the input is denoted by x, then the output is denoted by f (x) (read “f of x”). = The fundamental objects that we deal with in calculus are functions. = This chapter prepares the way for calculus by discussing: ◦ The basic ideas concerning functions ◦ Their graphs ◦ Ways of transforming and combining them = A function can be represented in 4 common ways: ◦ By an equation/formula ◦ In a table ◦ By a graph ◦ In words/verbal Mathematics I – Dr.Nobphadon Suksangpanya 2  FUNCTIONS – Examples The area (A) of a circle depends on the radius (r) of the circle. The rule that connects r and A is given by the equation A = π r 2 With each positive number r, there is associated one value of A, and we say that A is a function of r. The human population of the world (P) depends on the time (t).  The table gives estimates of the world population P(t) at time t, for certain years.  For instance, P(1950) ≈ 2,560, 000, 000  However, for each value of the time t, there is a corresponding value of P, and we say that P is a function of t. Each of these examples describes a rule whereby; “given a number (r or t), another number (A or P) is assigned.”  In each case, we say that the second number is a function of the first number. Mathematics I – Dr.Nobphadon Suksangpanya 3  DOMAIN & RANGE If x and y are related by the equation y = f(x), then the set of all allowable inputs (x-values) is called the domain of f, and the set of outputs (y-values) that result when x varies over the domain is called the range of f. VALUE & RANGE The number f(x) is the value of f at x and is read ‘f of x.’ The range of f is the set of all possible values of f(x) as x varies throughout the domain. Example: A table describes a functional relationship y = f (x). the domain is the set {0, 1, 2, 3} the range is the set {−1, 3, 4, 6} Mathematics I – Dr.Nobphadon Suksangpanya 4  INDEPENDENT & DEPENDENT VARIABLE If x and y are related by the equation y = f(x), the variable x is called the independent variable (or argument) of f and the variable y is called the dependent variable of f. Independent variable = A symbol that represents an arbitrary number in the domain of a function f is called an independent variable. * For instance, in Example 𝐴𝐴 = 𝜋𝜋𝑟𝑟 2 , r is the independent variable. Independent variable = A symbol that represents a number in the range of f is called a dependent variable. * For instance, in Example 𝐴𝐴 = 𝜋𝜋𝑟𝑟 2 , A is the dependent variable. Mathematics I – Dr.Nobphadon Suksangpanya 5  MACHINE It’s helpful to think of a function as a machine.  If x is in the domain of the function f, then when x enters the machine, it’s accepted as an input and the machine produces an output f(x) according to the rule of the function.  Thus, we can think of the domain as the set of all possible inputs and the range as the set of all possible outputs. Mathematics I – Dr.Nobphadon Suksangpanya 6  ARROW DIAGRAM Another way to picture a function is by an arrow diagram.  Each arrow connects an element of D to an element of E.  The arrow indicates that f(x) is associated with x, f(a) is associated with a, and so on. Mathematics I – Dr.Nobphadon Suksangpanya 7  GRAPH The most common method for visualizing a function is its graph.  If f is a function with domain D, then its graph is the set of ordered pairs {( x, f ( x)) | x ∈ D}  Notice that these are input-output pairs.  In other words, the graph of f consists of all points (x, y) in the coordinate plane such that y = f(x) and x is in the domain of f. The graph of a function f gives us a useful picture of the behavior or ‘life history’ of a function.  Since the y-coordinate of any point (x, y) on the graph is y = f(x), we can read the value of f(x) from the graph as being the height of the graph above the point x. Mathematics I – Dr.Nobphadon Suksangpanya 8  GRAPH The graph of f also allows us to picture:  The domain of f on the x-axis  Its range on the y-axis Mathematics I – Dr.Nobphadon Suksangpanya 9  GRAPH Example 1 The graph of a function f is shown. a. Find the values of f(1) and f(5). b. What is the domain and range of f ? Mathematics I – Dr.Nobphadon Suksangpanya 10  GRAPH Example 2 Sketch the graph and find the domain and range of each function. a. f(x) = 2x – 1 b. g(x) = x2 Mathematics I – Dr.Nobphadon Suksangpanya 11  PIECEWISE-DEFINED FUNCTIONS Function f can change, depending on the value of x A function f is defined by: For this particular function, the rule is: 1 − x if x ≤ 1  First, look at the value of the input x. f ( x)  2  If it happens that x≤1 , then the value of f(x) is 1 - x. x if x > 1  In contrast, if x > 1, then the value of f(x) is x2. Evaluate f(0), f(1), and f(2) and sketch the graph. Mathematics I – Dr.Nobphadon Suksangpanya 12  PIECEWISE-DEFINED FUNCTIONS The next example is the absolute value function.  Recall that the absolute value of a number a, denoted by |a|, is the distance from a to 0 on the real number line.  Distances are always positive or 0.  So, we have | a |≥ 0 for every number a.  For example, |3| = 3 , |-3| = 3 , |0| = 0 , | 2 − 1|= 2 − 1 , | 3 − π |= π − 3 In general, we have: |a|= a if a ≥ 0 | a | = −a if a < 0 Remember that, if a is negative, then -a is positive. Mathematics I – Dr.Nobphadon Suksangpanya 13  PIECEWISE-DEFINED FUNCTIONS Find a formula for the function f graphed in the figure. Mathematics I – Dr.Nobphadon Suksangpanya 14  SYMMETRY: EVEN FUNCTION If a function f satisfies f(-x) = f(x) for every number x in its domain, then f is called an even function.  For instance, the function f(x) = x2 is even because f(-x) = (-x)2 = x2 = f(x) The geometric significance of an even function is that its graph is symmetric with respect to the y–axis.  This means that, if we have plotted the graph of f for x≥0, we obtain the entire graph simply by reflecting this portion about the y-axis. Mathematics I – Dr.Nobphadon Suksangpanya 15  SYMMETRY: EVEN FUNCTION Mathematics I – Dr.Nobphadon Suksangpanya 16  SYMMETRY: ODD FUNCTION If f satisfies f(-x) = -f(x) for every number x in its domain, then f is called an odd function. ◦ For example, the function f(x) = x3 is odd because f(-x) = (-x)3 = -x3 = -f(x) The graph of an odd function is symmetric about the origin.  If we already have the graph of f for x≥0 , we can obtain the entire graph by rotating this portion through 180° about the origin. Mathematics I – Dr.Nobphadon Suksangpanya 17  SYMMETRY Example Determine whether each of these functions is even, odd, or neither even nor odd. f(x) = x5 + x f (− x) = (− x)5 + (− x) = (−1)5 x5 + (− x) =− x5 − x =−( x5 + x) = − f ( x) Thus, f is an odd function. Mathematics I – Dr.Nobphadon Suksangpanya 18  SYMMETRY Example Determine whether each of these functions is even, odd, or neither even nor odd. g(x) = 1 - x4 g (− x) = 1 − (− x 4 ) = 1 − x 4 = g ( x) So, g is even. Mathematics I – Dr.Nobphadon Suksangpanya 19  SYMMETRY Example Determine whether each of these functions is even, odd, or neither even nor odd. h(x) = 2x - x2 h(− x) =2(− x) − (− x) 2 =−2 x − x 2 Since h(-x) ≠ h(x) and h(-x) ≠ -h(x), we conclude that h is neither even nor odd. Mathematics I – Dr.Nobphadon Suksangpanya 20  SYMMETRY The graphs of the functions in the example are shown.  The graph of h is symmetric neither about the y-axis nor about the origin. Mathematics I – Dr.Nobphadon Suksangpanya 21  INCREASING AND DECREASING FUNCTIONS The function f is said to be This graph rises from A to B, increasing on the interval [a, b], falls from B to C, decreasing on [b, c], and rises again from C to D. and increasing again on [c, d]. Mathematics I – Dr.Nobphadon Suksangpanya 22  INCREASING AND DECREASING FUNCTIONS You can see from the figure that the function f(x) = x2 is decreasing on the interval (−∞, 0] and increasing on the interval [0, ∞) Mathematics I – Dr.Nobphadon Suksangpanya 23  Common Graphs The purpose of this section is to make sure that you’re familiar with the graphs of many of the basic functions that you’re liable to run across in a calculus class. Mathematics I – Dr.Nobphadon Suksangpanya 24 Mathematics I – Dr.Nobphadon Suksangpanya 25 Mathematics I – Dr.Nobphadon Suksangpanya Mathematics I – Dr.Nobphadon Suksangpanya 27 Mathematics I – Dr.Nobphadon Suksangpanya 28 Mathematics I – Dr.Nobphadon Suksangpanya 29 Mathematics I – Dr.Nobphadon Suksangpanya 30 Mathematics I – Dr.Nobphadon Suksangpanya 31 Mathematics I – Dr.Nobphadon Suksangpanya 32 Mathematics I – Dr.Nobphadon Suksangpanya 33 Mathematics I – Dr.Nobphadon Suksangpanya 34 Mathematics I – Dr.Nobphadon Suksangpanya 35 Mathematics I – Dr.Nobphadon Suksangpanya 36 Mathematics I – Dr.Nobphadon Suksangpanya 37  Limits Dr. Nobphadon Suksangpanya Mathematics I – Dr.Nobphadon Suksangpanya 1  Outline What a limit tells us about a function How the limit can be used to get the rate of change of a function How the limit can be used to get the slope of the line or tangent to the graph of a function (Note: we'll be seeing other, easier, ways of doing these later) Investigate limit properties How a variety of techniques to employ when attempting to compute a limit Look at limits whose "value" is infinity and how to compute limits at infinity Concept of continuity Mathematics I – Dr.Nobphadon Suksangpanya 2  Limit Definition Simpler definition: As x gets closer to x=a (from both sides), then f(x) must be getting closer and closer to L. As we move in towards x=a, then f(x) must be moving in towards L. Ex: The limit notations for the two previous problems are: Mathematics I – Dr.Nobphadon Suksangpanya 3  Steps to estimate limits: 1. Take x’s on both sides of x=a that move in closer and closer to a 2. Plug these values of x into the function. 3. Check if the function values are moving in towards a number and use this as our estimate. Example notice that we can’t plug in x=2 into the function as this would give us a division by zero error. This is not a problem since the limit doesn’t care what is happening at the point in question. Solution We will choose values of x that get closer and closer to x=2 and plug these values into the function. Doing this gives the following table of values. From this table it appears that the function is going to 4 as x approaches 2, so Mathematics I – Dr.Nobphadon Suksangpanya 4  Analyze the problem: A large open dot at x=2  function doesn’t exist at x=2 BUT as x moves in towards 2 (from both sides), the function is approaching y=4. Limits are asking what the function is doing around x=a and are NOT concerned with what the function is actually doing at x=a Mathematics I – Dr.Nobphadon Suksangpanya 5  Example Solution Note that now  moving toward x=2, the function is approaching a y = 4 means what the function is doing around x=2 and we don’t care what the function is doing at x=2 Therefore,  The limit is NOT 6! Mathematics I – Dr.Nobphadon Suksangpanya 6  Example Solution From this table, we would guess that the limit is 1. However, if we did make this guess, we would be wrong. Consider any of the following function evaluations. Mathematics I – Dr.Nobphadon Suksangpanya 7  When using a table of values, there will always be the possibility that we aren’t choosing the correct values. In fact, this is such a problem that after this section we will never use a table of values to guess the value of a limit again. Moving towards t=0, the function starts oscillating wildly. The oscillations increases in speed the closer to t=0 that we get. Recall from our definition of the limit that in order for a limit to exist the function must be settling down in towards a single value as we get closer to the point in question. This function clearly does not settle in towards a single number and so this limit does not exist! Mathematics I – Dr.Nobphadon Suksangpanya 8  Example Solution This function is often called either the Heaviside or step function. We could use a table of values to estimate the limit, but it’s probably just as quick in this case to use the graph so let’s do that. Below is the graph of this function. Approach t=0 from the right side, the function is moving in towards a y value of 1. Approach t=0 from the left side, the function is moving in towards a y value of 0. The function settle down to a single number as t=0 on either side. The problem is that the number is different on each side of t=0. “One–Sided Limits” Mathematics I – Dr.Nobphadon Suksangpanya 9  One–Sided Limits As the name implies, with one-sided limits we will only be looking at one side of the point in question. Here are the definitions for the one-sided limits. Mathematics I – Dr.Nobphadon Suksangpanya 10  Example Estimate the value of the following limits. Solution The function is moving in towards a value of 1 as we get closer to t=0 from the right side. The right-handed limit is The function is moving in towards a value of 0 as we get closer to t=0 from the left side. The left-handed limit is In this example we do get one-sided limits even though the normal limit itself doesn’t exist. Two one-sided limits both existed but did not have the same value. Normal limit did not exist. Mathematics I – Dr.Nobphadon Suksangpanya 11  Example Solution we can see that both of the one-sided limits suffer the same problem that the normal limit did in the previous section. The function does not settle down to a single number on either side of t=0. Therefore, neither the left-handed nor the right- handed limit will exist in this case. Mathematics I – Dr.Nobphadon Suksangpanya 12  Example Estimate the value of the following limits. Solution In this case regardless of which side of x=2 we are on the function is always approaching a value of 4 and so we get, one-sided limits & the normal limit all had a value of 4 Mathematics I – Dr.Nobphadon Suksangpanya 13  The relationship between one-sided limits and normal limits can be summarized by the following fact. Mathematics I – Dr.Nobphadon Suksangpanya 14  Exercise Given the following graph, Mathematics I – Dr.Nobphadon Suksangpanya 15  Computing Limits Strategy/techniques for finding exact value of limits. Limits Properties Mathematics I – Dr.Nobphadon Suksangpanya 16 Mathematics I – Dr.Nobphadon Suksangpanya 17 Mathematics I – Dr.Nobphadon Suksangpanya 18  Example Solution This is a combination of several of the functions listed above and none of the restrictions are violated so all we need to do is plug in x=3 into the function to get the limit. Mathematics I – Dr.Nobphadon Suksangpanya 19  ** However, there are also many limits for which the method won’t work. Example if we try to plug in x=2 we get 0/0 !!! Therefore, we need techniques for dealing with the limits that will not allow us to just use this fact. Note: is called “indeterminate form”. Can get Variety of answers! Indeterminate form = Cannot find solutions for some form of Mathematical expressions. In rare cases, limits of indeterminate forms can be found by using algebraic simplification. In most cases, need to use other methods  we will get there later on! Mathematics I – Dr.Nobphadon Suksangpanya 20  Example Evaluate the following limit. Solution Mathematics I – Dr.Nobphadon Suksangpanya 21  Example Evaluate the following limit. Solution Mathematics I – Dr.Nobphadon Suksangpanya 22  Example Evaluate the following limit. Solution Mathematics I – Dr.Nobphadon Suksangpanya 23  There is one more limit that we need to do. However, we will need a new fact about limits that will help us to do this. Mathematics I – Dr.Nobphadon Suksangpanya 24  Example Evaluate the following limit. Solution Mathematics I – Dr.Nobphadon Suksangpanya 25 Mathematics I – Dr.Nobphadon Suksangpanya 26  Infinite Limits Limits whose value is infinity or minus infinity. For these cases, the Infinite limit can be called “Normal Limit”. Mathematics I – Dr.Nobphadon Suksangpanya 27  Example Solution Mathematics I – Dr.Nobphadon Suksangpanya 28  Example Solution Mathematics I – Dr.Nobphadon Suksangpanya 29  Example Solution right-hand limit, as x gets closer to -2, then x+2 will be getting closer to zero left-hand limit, as x gets closer to -2, then x+2 will get closer to zero (and be negative) Mathematics I – Dr.Nobphadon Suksangpanya 30  Example Solution Mathematics I – Dr.Nobphadon Suksangpanya 31  Example Solution + + right-hand limit - - left-hand limit Mathematics I – Dr.Nobphadon Suksangpanya 32  Example A B Solution the normal limit will not exist because the two one-sided limits are not the same Mathematics I – Dr.Nobphadon Suksangpanya 33  More facts about infinite limits: Mathematics I – Dr.Nobphadon Suksangpanya 34  Limits at Infinity In the previous section we saw limits that were infinity and it’s now time to look at limits at infinity. By limits at infinity we mean one of the following two limits. Look at what happens to a function if we let x get very large in either the positive or negative Many of the limits that we’re going to be looking at we will need the following two facts. to avoid cases such as r=1/2 b/c taking square root of negative numbers = complex and we want to avoid that at this level Mathematics I – Dr.Nobphadon Suksangpanya 35  Example Evaluate each of the following limits. Solution We are probably tempted to say that the answer is zero (because we have an infinity minus an infinity) or maybe -∞ (because we’re subtracting two infinities off of one infinity). However, in both cases we’d be wrong. This is one of those indeterminate forms that we first started seeing in a previous section. Infinities just don’t always behave as real numbers do when it comes to arithmetic. Without more work there is simply no way to know what ∞-∞ will be. So we really need to be careful with this kind of problem. Mathematics I – Dr.Nobphadon Suksangpanya 36 Mathematics I – Dr.Nobphadon Suksangpanya 37 Mathematics I – Dr.Nobphadon Suksangpanya 38  Example Solution Just “plug” in the infinity This is another indeterminate form. In this case we might be tempted to say that the limit is infinity (because of the infinity in the numerator) zero (because of the infinity in the denominator) -1 (because something divided by itself is one) BUT all of these guesses are wrong !!! Mathematics I – Dr.Nobphadon Suksangpanya 39  Example Solution when we have a polynomial divided by a polynomial. We first identify the largest power of x in the denominator and we then factor this out of both the numerator and denominator. Doing this for the first limit gives cancel the x4 from both the numerator and the denominator and then use the Fact to take the limit of all the remaining terms. This gives the answer of -2/5 Mathematics I – Dr.Nobphadon Suksangpanya 40  Example Solution Mathematics I – Dr.Nobphadon Suksangpanya 41  Limits at Infinity: Exponential Functions Example Solution Mathematics I – Dr.Nobphadon Suksangpanya 42  The main point of the previous example was to point out that if the exponent goes to positive infinity in the limit, then the exponential function will go to infinity in the limit. if the exponent goes to minus infinity in the limit, then the exponential function will go to zero in the limit. In case of limit problems with sums and/or differences of exponential functions; if the limit is at +∞, we only look at exponentials with positive exponents if the limit is at - ∞, we only look at exponentials with negative exponents. Mathematics I – Dr.Nobphadon Suksangpanya 43  Sums and/or differences of exponential functions Example Solution Mathematics I – Dr.Nobphadon Suksangpanya 44  Sums and/or differences of exponential functions Example Solution Mathematics I – Dr.Nobphadon Suksangpanya 45  Rational functions involving exponentials. Example Solution Mathematics I – Dr.Nobphadon Suksangpanya 46  Rational functions involving exponentials. Example Solution Mathematics I – Dr.Nobphadon Suksangpanya 47  Continuity Over the last few sections we’ve been using the term “nice enough” to define those functions that we could evaluate limits by just evaluating the function at the point in question. It’s now time to formally define what we mean by “nice enough”. In other words, a function is continuous if its graph has no holes or breaks in it. Mathematics I – Dr.Nobphadon Suksangpanya 48  Example Given the graph of f(x), shown below, determine if f(x) is continuous at x=−2, x=0, and x=3. not continuous jump discontinuity jump discontinuity = graph has a break in it & values of the function to either side of the break are finite (i.e., the function doesn’t go to infinity) continuous not continuous removable discontinuity Mathematics I – Dr.Nobphadon Suksangpanya Removable discontinuity = there is a hole in the graph. 49  Example Determine where the function below is not continuous. Mathematics I – Dr.Nobphadon Suksangpanya 50  Example Evaluate the following limit. Since we know that exponentials are continuous everywhere we can use the fact above. Mathematics I – Dr.Nobphadon Suksangpanya 51  Example Solution Mathematics I – Dr.Nobphadon Suksangpanya 52  Applications of Limit: Tangent Lines Definition: A tangent line to the function f(x) at the point x=a is a linear line that just touches the graph of the function at the point in question and is “parallel” (in some way) to the graph at that point. The tangent line equation : 𝒚𝒚 = 𝒎𝒎𝒎𝒎 + 𝒄𝒄 Mathematics I – Dr.Nobphadon Suksangpanya 53  Example Find the tangent line to f(x) = 15−2x2 at x=1 We wanted the tangent line to f(x) at a point x=a. 1. Define point P to be on the tangent line at x=a : P=(a,f(a)) 2. Define point Q to be on the graph at any x : Q=(x,f(x)) 3. compute the slope of the line connecting P and Q as follows: 15−2𝑥𝑥 2 − 15−2 12 𝑥𝑥 2 −1 𝑚𝑚𝑃𝑃𝑃𝑃 = = −2 𝑥𝑥−1 𝑥𝑥−1 4. take values of x that get closer and closer to x=a 4.1. must look at x’s on both sides of x=a 4.2. use this list of values to estimate the slope of the tangent line, m Mathematics I – Dr.Nobphadon Suksangpanya 54  Slope : Example Find the tangent line to f(x) = 15−2x2 at x=1 15−2𝑥𝑥 2 − 15−2 12 𝑥𝑥 2 −1 𝑚𝑚𝑃𝑃𝑃𝑃 = = −2 𝑥𝑥−1 𝑥𝑥−1 𝑥𝑥 2 − 1 𝑚𝑚 = lim −2 = −2 lim 𝑥𝑥 + 1 = −4 𝑥𝑥→1 𝑥𝑥 − 1 𝑥𝑥→1 The tangent line equation : 𝑦𝑦 = −4𝑥𝑥 + 𝑐𝑐 At 𝑥𝑥 = 1 , 𝑦𝑦 = 𝑓𝑓 1 = 13 ; 13 = −4 1 + 𝑐𝑐 The tangent line equation : 𝑦𝑦 = −4𝑥𝑥 +17 𝑐𝑐 = 17 Mathematics I – Dr.Nobphadon Suksangpanya 55  Applications of Limit: Rates of Change Consider a function, f(x), that represents some quantity that varies as x varies. Examples : f(x) represents the amount of water in a holding tank after x minutes. f(x) is the distance traveled by a car after x hours. In both of these example we used x to represent time BUT x doesn’t have to represent time. Instantaneous rate of change or rate of change of f(x) at x=a = How fast f(x) is changing at x=a. Average Rate of Change Instantaneous 𝑓𝑓 𝑥𝑥 − 𝑓𝑓 𝑎𝑎 = lim or A.R.C. of f(x) at x=a and any x: rate of change 𝑥𝑥→𝑎𝑎 𝑥𝑥 − 𝑎𝑎 Mathematics I – Dr.Nobphadon Suksangpanya 56  Example Suppose that the amount of air in a balloon after t hours is given by Estimate the instantaneous rate of change of the volume after 5 hours. Solution The first thing that we need to do is get a formula for the average rate of change of the volume. In this case this is, To estimate the instantaneous rate of change of the volume at t=5 we just need to pick values of t that are getting closer and closer to t=5. 𝑡𝑡 3 − 6𝑡𝑡 2 + 25 𝑡𝑡 − 5 𝑡𝑡 2 − 𝑡𝑡 − 5 lim = lim = lim 𝑡𝑡 2 − 𝑡𝑡 − 5 = 15 𝑡𝑡→5 𝑡𝑡 − 5 𝑡𝑡→5 𝑡𝑡 − 5 𝑡𝑡→5 Mathematics I – Dr.Nobphadon Suksangpanya 57  Application in Velocity Problem f(t) = function of an object position at time t. Instantaneous velocity of the object = rate at which the position is changing. To estimate the instantaneous velocity, we would first compute the average velocity, and then take values of t closer and closer to t=a and use these values to estimate the instantaneous velocity. Instantaneous 𝑓𝑓 𝑡𝑡 − 𝑓𝑓 𝑎𝑎 = lim velocity 𝑡𝑡→𝑎𝑎 𝑡𝑡 − 𝑎𝑎 Mathematics I – Dr.Nobphadon Suksangpanya 58  Derivatives Dr. Nobphadon Suksangpanya Mathematics I – Dr.Nobphadon Suksangpanya 1  Chapter 3 : Derivatives Section 3-1 : The Definition Of The Derivative Derivative means the rate of change of a function with respect to a variable We often “read” f’(x) as “f prime of x” Let’s compute a couple of derivatives using the definition. Mathematics I – Dr.Nobphadon Suksangpanya 2  Alternate Notation Evaluate the derivative at x=a Mathematics I – Dr.Nobphadon Suksangpanya 3  Example Solution Mathematics I – Dr.Nobphadon Suksangpanya 4  Example Solution Mathematics I – Dr.Nobphadon Suksangpanya 5  - Derivatives will not always exist. - Derivative of the absolute value function exists at every point except at x = 0. Mathematics I – Dr.Nobphadon Suksangpanya 6  Differentiation Formulas Previously used the definition of derivation involving limits and functions. Simplify derivative computation by using differentiation formulas. Mathematics I – Dr.Nobphadon Suksangpanya 7  Derivative of a Constant Function Example Mathematics I – Dr.Nobphadon Suksangpanya 8  The Power Rule If n is any real number, then Example Differentiate Mathematics I – Dr.Nobphadon Suksangpanya 9  The Constant Multiple Rule If c is a constant and f is a differentiable function, then Example 𝑑𝑑 3 (𝑐𝑐) 8 𝑥𝑥 5 = 𝑑𝑑𝑑𝑑 Mathematics I – Dr.Nobphadon Suksangpanya 10  The Sum Rule and the Difference Rule If f and g are both differentiable, then Example Mathematics I – Dr.Nobphadon Suksangpanya 11  The Product Rule If f and g are both differentiable, then Example (a) Mathematics I – Dr.Nobphadon Suksangpanya 12  Example Mathematics I – Dr.Nobphadon Suksangpanya 13  The Quotient Rule If f and g are both differentiable, then Example Differentiate the function Mathematics I – Dr.Nobphadon Suksangpanya 14  Example Differentiate the function Mathematics I – Dr.Nobphadon Suksangpanya 15  NOTE: Don’t use the Quotient Rule every time you see a quotient. Mathematics I – Dr.Nobphadon Suksangpanya 16  More Examples: Differentiate the functions 3) 1) 2) Mathematics I – Dr.Nobphadon Suksangpanya 17  Example: Find equations of the tangent line and normal line to the curve 𝑦𝑦 = 𝑥𝑥 𝑥𝑥 at 𝑥𝑥 = 1 Normal line Tanget line 1 𝑚𝑚normal = − 𝑚𝑚tangent Mathematics I – Dr.Nobphadon Suksangpanya 18 Mathematics I – Dr.Nobphadon Suksangpanya 19  Derivatives of Trigonometry Functions Mathematics I – Dr.Nobphadon Suksangpanya 20  Example Differentiate Differentiate Mathematics I – Dr.Nobphadon Suksangpanya 21  Example Differentiate Differentiate Mathematics I – Dr.Nobphadon Suksangpanya 22  Derivatives of Exponential and Logarithm Functions Example Differentiate each of the following functions. Mathematics I – Dr.Nobphadon Suksangpanya 23 Mathematics I – Dr.Nobphadon Suksangpanya 24  Chain Rule How to solve these functions: Mathematics I – Dr.Nobphadon Suksangpanya 25  Example differentiate 𝑦𝑦 = 5𝑥𝑥 − 8 Assume 𝑢𝑢 𝑥𝑥 = 5𝑥𝑥 − 8 , then 𝑦𝑦 𝑢𝑢 = 𝑢𝑢 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑢𝑢 Chain Rule: = 𝑑𝑑𝑑𝑑 𝑑𝑑𝑢𝑢 𝑑𝑑𝑑𝑑 Mathematics I – Dr.Nobphadon Suksangpanya 26  Example Differentiate each of the following 𝑓𝑓 𝑥𝑥 = 𝑒𝑒 4𝑥𝑥+1 𝑓𝑓 𝑥𝑥 = (2 − 4𝑥𝑥 2 )3 Mathematics I – Dr.Nobphadon Suksangpanya 27 Mathematics I – Dr.Nobphadon Suksangpanya 28 Mathematics I – Dr.Nobphadon Suksangpanya 29 Mathematics I – Dr.Nobphadon Suksangpanya 30 Mathematics I – Dr.Nobphadon Suksangpanya 31 Mathematics I – Dr.Nobphadon Suksangpanya 32 Mathematics I – Dr.Nobphadon Suksangpanya 33 Mathematics I – Dr.Nobphadon Suksangpanya 34 Mathematics I – Dr.Nobphadon Suksangpanya 35  Implicit Differentiation To this point we’ve done quite a few derivatives, but they have all been derivatives of functions of the form y = f(x). Unfortunately, not all the functions that we’re going to look at will fall into this form. Let’s take a look at an example of a function like this. Example Mathematics I – Dr.Nobphadon Suksangpanya 36  Example Mathematics I – Dr.Nobphadon Suksangpanya 37  Example Mathematics I – Dr.Nobphadon Suksangpanya 38  Example Mathematics I – Dr.Nobphadon Suksangpanya 39  Higher Order Derivatives Mathematics I – Dr.Nobphadon Suksangpanya 40 Mathematics I – Dr.Nobphadon Suksangpanya 41  Example Mathematics I – Dr.Nobphadon Suksangpanya 42  Example Mathematics I – Dr.Nobphadon Suksangpanya 43  Logarithmic Differentiation Taking the derivatives of some complicated functions can be simplified by using logarithms. This is called logarithmic differentiation. Example Mathematics I – Dr.Nobphadon Suksangpanya 44 Mathematics I – Dr.Nobphadon Suksangpanya 45  Example Mathematics I – Dr.Nobphadon Suksangpanya 46  Example Mathematics I – Dr.Nobphadon Suksangpanya 47  L’Hospital`s Rule and Indeterminate Forms So, L’Hospital’s Rule tells us that if we have an indeterminate form 0/0 or ∞/∞ all we need to do is differentiate the numerator and differentiate the denominator and then take the limit. Mathematics I – Dr.Nobphadon Suksangpanya 48  L’Hospital`s Rule and Indeterminate Forms L’Hospital’s Rule or Do NOT Use L’Hospital’s Rule Algebraic Simplification Mathematics I – Dr.Nobphadon Suksangpanya 49  L’Hospital`s Rule and Indeterminate Forms Example Evaluate each of the following limits. Mathematics I – Dr.Nobphadon Suksangpanya 50  Example Evaluate the following limit Mathematics I – Dr.Nobphadon Suksangpanya 51  Example Evaluate the following limit. Option #1 Option #2 Mathematics I – Dr.Nobphadon Suksangpanya 52  Example Evaluate the following limit. Mathematics I – Dr.Nobphadon Suksangpanya 53