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1 Section.1 Etrema on an Interval 1. Understand the definition of etrema of a function on an interval.. Understand the definition of relative etrema of a function on an open interval.. Find etrema on a closed interval. I. Etrema of a Function A. Absolute Etrema Theorem: The Etreme Value Theorem If f is on a closed interval [a,b], then f must have both a and a on the interval. B. Relative Etrema (or local etrema) 5 (,5) 5 (,5) 5 (,5) 5 (,5) (-1,) (0,1) 1 (-1,) (-1,) [ 1,] (0,1) 1 [ 1,] (0,1) 1 [ 1,] (0,1) 1 ( 1, ) (-1,) II Relative Etrema and Critical Numbers A. Def Let f be defined at c. If f is not at c or f ( c) 0, then c is a number of f. B. Relative Etrema occur only at Critical Numbers, if they occur at all. Theorem: If f has a relative etreme at c, then c is a critical number of f. C. Eamples: Find the etrema on the given closed intervals y on the interval 1, 1. [ ]. f ( ) + on the interval [-1,]. f ( ) sin cos on the interval [ 0,π ] Homework: p all, 11, 1, 15, 17, 19,, 7, 9,, 1,, 5-60 all, 61, all
2 Section. Rolle s Theorem and The Mean Value Theorem 1. Understand and use Rolle s Theorem.. Understand and use the Mean Value Theorem. I. Rolle s Theorem A. Rolle s Theorem Let f be on the closed interval [a,b] and on the open interval (a,b). If f ( a) f ( b), then there is number c in (a,b) such that f ( c) 0 Relative Maimum Relative Maimum d d d a c b a c b a B. Eample: Does Rolle s Theorem apply to the following functions on the given interval? If so, find all the c values such that f ( c) 0 1. f ( ) on the interval (,) b 1. f ( ) on the interval ( 1,5 ). f ( ) on the interval ( 0,5) II. The Mean Value Theorem for Derivatives. A. Theorem: 1. The Mean Value Theorem for Derivatives If f is on a closed interval [a,b] and on the open interval (a,b), then there a number c such that f ( c). (i.e.) MVT That the slope at a in the interval must equal the slope over the given.
3 (Instant Velocity) Tangent Line Slope of Secant Line Equals Slope of Tangent Line Secant Line (Avg Velocity) B. Eample 1. Does the MVT apply to the following function on the given interval? If so, find c f ( b) f ( a) such that f ( c). b a f ( ) 5 on the interval ( 1, ). Two stationary patrol cars equipped with radar are 5 miles apart on a highway. As a vehicle passes the first patrol car, its speed is clocked at 70 mph. Four minute later, when it crosses the second patrol car it is clocked at 65 mph. Did the vehicle eceed the 70 mph speed limit during those minutes? Homework: p. 17 1,,, 5, 11, 1, 15, 19, 1,, 5, 9, 1,, 7, 9,, 9, 50, 51, 5, all, 71, 7-80 all
4 Section. Increasing and Decreasing Functions and the First Derivative 1. To determine the intervals where a function increases or decreases. I. Increasing and Decreasing Functions A. Tests Let f be a function that is on the interval [a,b] and on the open interval (a,b). 1. If f ( ) > 0, then f is.. If f ( ) < 0, then f is.. If f ( ) 0, then f is. Decreasing Increasing B. Eample: 1. Find the intervals on which f ( ) Constant is increasing or decreasing.. Method a) Locate CP s b) Pick a test value. c) Make conclusion. II. The First Derivative Test A. Test: 1. If f switches from CP is a maimum.. If f switches from CP is a minimum.. If f does not switch signs at CP CP is neither, but a Point of Inflection. B. Eamples Find the relative etrema for the following functions on the given interval and verify on your calculator 1. f ( ) 1 sin on the interval ( 0,π ). ( ) ( ) f. f ( ) + 1 Homework: p ,, 6, 1, 15, 1, 6, 9, 1,, 9,, all, all, (79, 80) b and c only
5 Section. Concavity and Second Derivative Test 1. Determine intervals on which a function is concave upward or concave downward.. Find any points of inflection of the graph of a function. f (). Apply the Second Derivative Test to find relative etrema of a function. I. Concavity A. Definition Let f be on open interval I. The graph of f is concave on I if f is on the interval and concave of I if f is on the interval. Eplain how this works f ' () B. Test for Concavity 1. If y < 0, then concave.. If y > 0, then concave. C. Eample: y + 0 f '' () II. Points of Inflection A. Point of Inflection 1. A point where the changes.. Requires that the tangent line eists at that point.. The graph crosses its tangent line at a P of I. NOT A P of I ( ). Can only occur at c, f ( c ) when f ( c) 0 or when f ( c) does not eist.
6 B. Eample Find the points of inflection, if they eist, for the following functions. 1. f ( ). f ( ) III. Second Derivative Test A. Test Let f be a function such that f ( c) 0 and the second derivative of f eists on the open interval containing c. 1. If f ( c) > 0, then f ( c ) is a relative.. If f ( c) < 0, then f ( c ) is a relative.. If f ( c) 0, then the test fails. Use First Derivative Test. B. Eample Use the nd Derivative Test to find the relative etrema f ( ) + 5. f ( ) + 1 Homework: p all, 5, 11, 15, 1,, 7,, 9,, 9, 51, 57-6 all, 65, 67, 70, 7, 91-9 all
7 Section.5 Limits at infinity 1. To find the value of a function (end behavior) when approaches infinity.. To demonstrate what it means for a function to have a limit as approaches infinity.. To find horizontal asymptotes. I. Limits at Infinity (End Behavior) Suppose you were to sketch the graph of the function f given by f ( ),but you needed to + 1 know what was happening to the graph at the left most and right most points. To get an idea of the end behavior we will develop three methods. A. Numerically decreases without bound. increases without bound f() 0 f() approaches. f() approaches. B. Graphically When you graph the function on your calculator, what does it appear to be doing? C. Analytically Rule: lim ± c n D. Eamples: 1. lim Short Cuts (From Lab #) lim 9 6.
8 E. Eception to the Rule Functions that are even indeed radicals 1. lim lim + 1 F. More Eamples 1. lim sin A problem occurs when we divide inside the radical in the denominator by. Why? How can we take care of it?. sin lim Algebraically: 1 y 1/ π π π π 5π y (sin )/ y -1/ 1 Graphically II. Horizontal Asymptotes A. If lim f ( ) b or lim f ( ) b, then y b is the horizontal asymptote. B. Eamples: Find the horizontal asymptotes, if any. 1. f ( ). f ( ) f ( ) III. Infinite Limits at Infinity Eamples: 1. lim + 1. lim + 1 Homework: Day 1: p all, 1, 1,, 5, 9, 1, 8, 55, 107, 108 Day : Calculator Lab #8 Day : p all, 5-19 odds, 61-6 all, 67, 71
9 Section.7 Optimization Problems 1. Solve applied minimum and maimum problems. I. Optimization Problems A. Eample 1. You have been asked to design a one-liter oilcan shaped like a right circular cylinder. What dimensions will use the least material? (1L 1000 cubic cm) Main Eq: Secondary Eq:
10 . Two posts, one 1 feet high and the other 8 feet high, stands 0 feet apart. They are to be stayed by two wires, attached to a single stake, running from ground level to the top of each post. Where should the stake be placed to use the least wire? Primary Eq: Secondary Eq: Homework: p. 17, 1,, 6,, 9, 56
11 Section.9 Differentials 1. Understand the concept of a tangent line approimation.. Compare the value of the differential, dy, with the actual change in y, y.. Estimate a propagated error using a differential.. Find the differential of a function using differentiation formulas. I. Tangent Line Approimation (Linear Approimation) A. The approimation f() f(a) + f (a)( a) is called the linear approimation or tangent line approimation of f at a. The equation of this tangent line is y f(a) f (a)( a) By zooming in toward a point on the graph of a differentiable function, we notice that the graph looks more and more like its tangent line. B. Eample 1. Consider the curve defined by dy a) Find. d y y 19 b) Write an equation for the line tangent to the curve at the point (,-1) c) There is a number k so that the point (., k) is on the curve. Using the tangent line found in part (b), approimate the value of k. d) Find the actual value of k on the curve. What was the percent of error?
12 . Use the given graph to approimate a) f (1.9) f ' b) f (.0) 1 (,1) f II. Differentials Eamples Function Derivative Differential 1. y 1 5. y sin. y cos. 1 y III. Error Propagation A. differential dy Relativer Error actual y V B. Eample 1. The measured radius of a ball bearing is 0.7 inch. If the measurement is correct within 0.01 inch, estimate the propagated error in the volume V of the ball bearing. π r. Find the relative error and percent of error. Homework: p. 0 1, 11, 1, 1, 18,,, 5, 7, 0,
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