Presentation on theme: "Click to see more. Rate of Change And Limits What is Calculus?"— Presentation transcript:
1Click to see more.Rate of ChangeAnd LimitsWhat is Calculus?
2Two Basic Problems of Calculus 1. Find the slope of the curve y = f (x) at the point (x, f (x))(x, f(x))(x, f(x))The slope problem is the first major problem of calculus. You already know how to find the slope of a line. But how can you find the slope at a point on a curve. As you can see, the slope of a curve can change from one point to another. There are many applications to the slope problem, since a slope indicates a rate of change.(x, f(x))
32. Find the area of the region bounded above by the curve y = f(x), below by the x-axis and by the vertical lines x = a and x = bAreay = f(x)abxThe area problem is the second major problem of calculus. There are many applications.
4From BC (before calculus) We can calculate the slope of a line given two pointsCalculate the slope of the line between the given point P (.5, .5) and another point on the curve, say Q(.1, .99). The line is called a secant line..
5Slope of Secant line PQ x f(x) .1 .99 .2 .98 .3 .92 .4 .76 P(0.5, 0.5) Point Qxf(x).18.104.22.168.22.214.171.124Let x values get closer and closer to .5. Determine f(x) values.
6Slope of Secant line PQ As Q gets closer to P, the Slope of the secant line PQGets closer and closer tothe slopeOf the line tangent to theCurve at P.
7Slope of a curve at a point Figure 1.4: The tangent line at point P has the same steepness (slope) that the curve has at P.The slope of the curve at a point P is defined to be the slope of the line that is tangent to the curve at point P.In the figure the point is P(0.5, 0.5)The slope of the curve at P(0.5, 0.5) is defined to be the slope of the line that is tangent to the curve at point P.
8Slope formulaIn calculus we learn how to calculate the slope at a given point P. The strategy is to take use secant lines with a second point Q. and find the slope of the secant line.Continue by choosing second points Q that are closer and closer to the given point P and see if the difference quotient gets closer to some fixed value..
9SlopeFind the slope of y = x2 at the point (1,1) Find the equation of the tangent line.A
10Find slope of tangent line on f(x) =x2 at the point (1,1) Approaching x = 1 from the rightxf(x)Slope of secant between (1,1) and (x, f(x))24126.96.36.199.11.212.11.011.0212.011.0012.001Slope appears to be getting close to 2.
11Find slope of tangent line on f(x) =x2 at the point (1,1) Approaching x = 1 from the leftxf(x)Slope of secant between (1,1) and (x, f(x))188.8.131.52.9.8184.108.40.206011.99.9991.999Slope appears to be getting close to 2.
12Write the equation of tangent line As the x value of the second point gets closer and closer to 1, the slope gets closer and closer to 2. We say the limit of the slopes of the secant is 2. This is the slope of the tangent line.To write the equation of the tangent line use the point-slope formula
13Average rate of change (from bc) If f(t) represents the position of an object as a function of time, then the rate of change is the velocity of the object.Find the average velocity if f (t) = 2 + cost on [0, ]1. Calculate the function value (position) at each endpoint of the intervalf() = 2 + cos () = 2 – 1 = 1f(0) = 2 + cos (0) = = 32. Use the slope formulaThe average velocity on on [0, ] is
14Instantaneous rate of change To calculate the instantaneous rate of change of we could not use the slope formula since we do not have two points.To approximate instantaneous calculate the average rates of change in shorter and shorter intervals to approximate the instantaneous rate of change.
152.2To understand the instantaneous rate of change (slope) problem and the area problem, you will need to learn about limitsThis important concept ties the ideas of calculus together.
16The answer can be found graphically, numerically and analytically. LimitsWhat happens to the value of f (x) when the value of x gets closer and closer and closer (but not necessarily equal) to 2?We write this as:Read as the limit of f(x) as x approaches 2.The answer can be found graphically, numerically and analytically.
17What happens to f(x) as x gets closer to 2? Graphical Analysis5432168101214161820f (x)xInput the function to Y1 in your graphing calculator. The “hole” is greatly exaggerated in this slide but can not be seen with the calculator.What happens to f(x) as x gets closer to 2?
18Numerical Analysis Use one sided limits Start to the left of 2 and choose x values getting closer and closer (but not equal) to 2x220.127.116.111.9991.9999f (x)9.25You will have to approach 2 from the left (numbers less than 2) and from the right (numbers greater than 2). Use your graphing calculator table function. Set the table so the independent variable is on ASK. That way you can select x and the calculator will return a value for f(x). See what happens if you try to ask for the value at x = 2.11.4111.941Could x get closer to 2? Does f(x) appear to get closer to a fixed number?
19Numerical AnalysisStart to the right of 2 and choose x values getting closer and closer (but not equal ) to 2x2.52.12.012.0012.0001f (x)15.312.61Use your graphing calculator table function. Set the table so the independent variable is on ASK. That way you can select x and the calculator will return a value for f(x). See what happens if you try to ask for the value at x = 2.If the limit exists, f(x) must approach the same value from both directions. Does the limit exist? Guess what it is.
20Limits that do not exist Figure 1.8: The functions in Example 7.Limits that do not existIn order for a limit to exist, the function must approach the same valueFrom the left and from the right.
21Infinite LimitsWhat happens to the function value as x gets closer and closer to 3 from the right?x3.53.13.013.0013.0001y311101100110001100001The function increases without bound so we sayThere is a vertical asymptote at x = 3.
22The line x=a is a Vertical Asymptote if at least one is true. Identify any vertical asymptotes:
23Graph of f(x)True or falsex = 2 is in the domain of f(b)(c)
242.3 Functions That Agree at All But One Point If f(x) = g(x) for all x in an open interval except x = c then:ExamplethenA long as the limit in the denominator is not equal to 0, you can use direct substitution here.Evaluate by direct substitution 2-5 = -3As x gets closer and closer and closer to 2, the function value gets closer and closer to -3.
25Analytic = = As x gets closer and closer to 2 (but not equal to 2) Using direct substitution,What are the two functions that agree in value with the exception of at a =2? What about the limits as x approaches 2?As x gets closer and closer to 2 (but not equal to 2)f(x) gets closer and closer to 12
27Basic LimitsIf b and c are real numbers and is n a positive integer1.Guess an answer and click to check.Guess an answer and click to check.Guess an answer and click to check.= -2Ex:2.= 5Ex:Try to see how to apply basic limits.3.= 9Ex:
28Properties of Limits Multiplication by a constant b Limit of a sum or differenceLimit of a productLimit of a powerThese properties look complicated but in practice are easy to use. Don’t spend too much time analyzing them. They are presented here to justify the use of direct substitution in many functions.Limit of a quotientwhen denominatoris not 0.
29Using Properties of Limits Properties allow evaluation of limits by direct substitution for many functions.Ex.:Since the limit in the denominator is not equal to 0 (it is 3), you can use direct substitution to evaluate the limit.As x gets closer and closer to 3, the function value gets closer and closer to 9.
30Direct substitution Analytic Techniques First substitute the value of x being approached into the function f(x). If this is a real number then the limit is that number.If the function is piecewise defined, you must perform the substitution from both sides of x. The limit exists if both sides yield the same value. If different values are produced, we say the limit does not exist.
31Rationalize a numerator or denominator Simplify a complex fraction Analytic TechniquesRewrite algebraically if direct substitution produces an indeterminate form such as 0/0Factor and reduceRationalize a numerator or denominatorSimplify a complex fractionFor analytic techniques you will need a good algebraic background.When you rewrite you are often producing another function that agrees with the original in all but one point. When this happens the limits at that point are equal.
32Find the indicated limit direct substitution failsRewrite and cancel= - 5now use direct sub.
33Find the indicated limit direct substitution failsRewrite and cancelnow use direct sub.
34Find the indicated limit 5calculate one sided limits7Since the one-sided limits are not equal, we say the limit does not exist. There will be a jump in the graph at x =2
35Determine the limit on y = sin θ/θ as θ approaches 0. Figure 1.24: The graph of f () = (sin )/.Determine the limit on y = sin θ/θ as θ approaches 0.Although the function is not defined at θ =0, the limit as θ is 1.
36A one-sided limitFigure 1.37: The graph of y = e1/x for x < 0 shows limx0– e1/x = 0. (Example 11)
37Limits that are infinite (y increases without bound) An infinite limit will exist as x approaches a finite value when direct substitution producesIf an infinite limit occurs at x = c we have a vertical asymptote with the equation x = c.
382.5 Continuity in (a) at x = 0 but not in other graphs. Figure 1.50: The function in (a) is continuous at x = 0; the functions in (b) through ( f ) are not.
39Conditions for continuity A function y = f(x) is continuous at x = c if and only if:The function is defined at x = cThe limit as x approaches c existsThe value of the function and the value of the limit are equal.
40Find the reasons for discontinuity in b, c, d, e and f. (a) Is continuous. (b) f(0) is not defined c) the value of the function does not equal the value of the limit (d) the limit as x approaches 0 does not exist (e) and (f) f(0) is not defined
41Figure 1.53: Composites of continuous functions are continuous. Composite FunctionsIf two functions are continuous at x = c then their composition will be continuous.is continuous for all reals.Example:
42Exploring ContinuityAre there values of c and m that make the function continuousAt x = 1? Find c and m or tell why they do not exist.
442.6 Slope of secant line and slope of tangent line
45s(t) = 8(t3 – 6 t2 +12t)t s566472Position of a car at t hours.1. Draw a graph.2. Does the car ever stop?3. What is the average velocity for the following intervalsa. [0, 2],b. [.5, 1.5]c. [.9,1.1]4. Estimate the instantaneous velocity at t = 1
46s(t) = 8(t3 – 6 t2 +12t) 2. Appears to stop at t =2. (Velocity= 0) 3. What is the average velocity for[0, 2],[.5, 1.5][.9,1.1]ts(t)262.5371.563.953.3521.158.168a) 31 mphb) 26 mphc) mph
47Find an equation of the tangent line to y = 2x3 – 4 at the point P(2, 12) So, m = 24. Use the point slope form to write the equation
48Slope of the tangent line at x= a Figure 1.62: The tangent slope isf (x0 + h) – f (x0)hSlope of the tangent line at x= alimh0Q(a + h, f (a + h))f(a+h) – f(a)The letter h represents the change in x which is sometimes denoted as Δx. Notice this difference quotient is the change in y over the change in x.P(a, f(a))a + ha
49Other form for Slope of secant line of tangent line Let h = x - aThen x = a + h
50Find an equation of the tangent line at (3, ½) to At a = 3, m = - 1/8Using the point-slope formula: