4.1 Tangent Lines. y 2 y 1 = y 2 y 1

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1 41 Tangent Lines Introduction Recall tat te slope of a line tells us ow fast te line rises or falls Given distinct points (x 1, y 1 ) and (x 2, y 2 ), te slope of te line troug tese two points is cange in y cange in x = rise run = y 2 y 1, x 2 x 1 providing tat x 2 x 1 If x 2 = x 1, te line is vertical, and te slope does not exist For given points (x 1, y 1 ) and (x 2, y 2 ) satisfying te additional requirement tat x 2 x 1 = 1, te slope of te line becomes y 2 y 1 = y 2 y 1 x 2 x 1 1 Tis simple observation gives an important interpretation of te slope of a line: it is a number tat tells te vertical cange per (positive) unit orizontal cange wen traveling from point to point on te line For example, te lines sown below ave (from left to rigt) slopes 5, 4, and 1 2 Wen traveling along a line from left to rigt: lines wit large positive slopes are steep upills ; lines wit small positive slopes are gradual upills ; lines wit large negative slopes are steep downills ; and lines wit small negative slopes are gradual downills 182

2 copyrigt Dr Carol JV Fiser Burns ttp://wwwonematematicalcatorg 183 EXERCISE 1 1 Prove tat y 2 y 1 x 2 x 1 = y 1 y 2 x 1 x 2 Terefore, te order tat te points are listed wen calculating te slope of a line is unimportant 2 A line as slope 3 If te x-values of two points on te line differ by 1, ow muc do teir y-values differ by? If te x-values of two points differ by 2, ow muc do teir y-values differ by? 3 On te same grap, sketc lines tat ave slopes 1, 10, and On te same grap, sketc lines tat ave slopes 1, 10, and 1 10 tangent lines; informal discussion Te tangent line to a grap at a point P is te line tat best approximates te grap at tat point In oter words, it is te best linear approximation at P Tangent lines may or may not exist, as illustrated below Wen tey do exist, it is intuitively clear ow tey sould be drawn finding te slopes of tangent lines GOAL: Find te slope of te tangent line to te grap of a function f at te point (x, f(x)) PROBLEM: Two points are needed to find te slope of a line! To remedy tis problem, coose a second point tat is close to (x, f(x)), and find te slope of te line troug tese two points Wen te second point is very close to (x, f(x)), tis line sould be a good approximation to te tangent line Let denote some small number, positive or negative (Tink of as being, say, 01, 0001 or 001) Ten, te point (x +, f(x + )) is close to (x, f(x)) If > 0, te new point is to te rigt of (x, f(x)) If < 0, te new point is to te left of (x, f(x))

3 184 copyrigt Dr Carol JV Fiser Burns ttp://wwwonematematicalcatorg secant line Te line troug tese two points (x, f(x)) and (x+, f(x+)) is called a secant line It serves as an approximation to te desired tangent line In general, te closer te second point (x +, f(x + )) is to te initial point (x, f(x)), te better te approximation Te slope of te secant line troug te points (x, f(x)) and (x +, f(x + )) is: slope of secant line = = (x + ) x difference quotient let 0 Te quantity obtained above is called a difference quotient It represents te slope of te secant line troug te points (x, f(x)) and (x +, f(x + )) Since we expect te slope of te secant line to better approximate te slope of te tangent line as te second point moves closer to te first (wic appens as approaces 0), it is natural to investigate te it Tis it may or may not exist If it does exist, ten tere is a tangent line to te grap of f at te point (x, f(x)), and te it value gives te slope of te tangent line to te grap of f at te point (x, f(x)) Tis result is summarized next DEFINITION slope of te tangent line to te grap of f at te point (x, f(x)) If te it m = 0 exists, ten tere is a nonvertical tangent line to te grap of f at te point (x, f(x)), and te number m gives te slope of tis tangent line investigating te it; wat are x and? Te it uses two letters, x and Te letter is te dummy variable for te it; it merely represents a number tat is getting arbitrarily close to zero Te it can equally well be written wit a different dummy variable, say f(x + x) f(x) x 0 x or f(x + t) f(x) t 0 t (Te symbol x is read as delta x, and denotes a cange in x)

4 copyrigt Dr Carol JV Fiser Burns ttp://wwwonematematicalcatorg 185 Te letter x tat appears in te it is te x-value of te point were te slope of te tangent line is desired If, for example, te slope is desired at te point (2, f(2)), ten te it becomes f(2 + ) f(2) f(x+) f(x) Note tat te it can only be investigated at a value of x were f is defined, so tat f(x) makes sense te it is a 0 0 situation EXAMPLE using te it formula to find te slopes of tangent lines Observe tat direct substitution of = 0 into te it yields a 0 0 situation Terefore, tis it can never be evaluated directly It is necessary to get f(x+) f(x) into a form tat displays wat is appening wen is close to zero, but not equal to zero In many cases, one tries to simplify te difference quotient to a point were tere is a factor of in te numerator, tat can be cancelled wit te in te denominator It s always best to test a new result in a situation were you already know te answer So, let s work first wit te function f(x) = 3x Te grap of f is a line of slope 3 If P is any point on tis line, ten te tangent line at P is te line itself, and we sould find tat te slope of te tangent line is 3 Let s see if te above formula bears tis out Let te first point be (x, f(x)) = (x, 3x), and let te second point be (x +, f(x + )) = (x +, 3(x + )) Te slope of te secant line between tese two points is and tus = 3(x + ) 3x = 3, 3 = = 3 Tus, for any point (x, f(x)) on te grap, te slope of te tangent line is 3, as expected EXERCISE 2 1 Consider te function f(x) = 3x Using te it formula, find te slope of te tangent line at te point (1, 3) 2 Consider te function f(x) = 3x Using te it formula, find te slope of te tangent line at a typical point (x, f(x)) 3 Consider te function f(x) = kx, were k is a nonzero constant Using te it formula, find te slope of te tangent line at a typical point (x, f(x)) 4 Consider te zero function f(x) = 0 Using te it formula, find te slope of te tangent line at a typical point (x, f(x))

5 186 copyrigt Dr Carol JV Fiser Burns ttp://wwwonematematicalcatorg EXAMPLE using te it formula to find te slopes of tangent lines Next, consider te function f(x) = x 2 + 2, wit grap sown below We expect to find tat: te slope of te tangent line at x = 0 is 0; wen x is small and positive, te slopes are small and negative; wen x is a large negative number, te slopes are large and positive Let s see if tis is borne out Here, f(x + ) = (x + ) 2 + 2, and we get: ( (x + ) ) ( x 2 + 2) = 0 (x 2 + 2x + 2 ) x 2 2 = x 2 2x 2 + x 2 = ( 2x ) = = ( 2x ) 0 = 2x Observe tat tis is a complete matematical sentence For a particular value of x, te = signs denote equality of real numbers Do NOT drop te it instruction until you actually let go to 0 Tis sentence sows tat te it exists for every value of x, and is equal to 2x Tat is, te slope of te tangent line at a point (x, f(x)) is 2x Te expected results are obtained: Wen x = 0, te slope of te tangent to te point (0, 2) is 2(0) = 0, as expected Wen x = 1, te slope of te tangent line to te point (1, 199) is 2(1) = 2, a small negative number, as expected Wen x = 4, te slope of te tangent line to te point ( 4, 14) is 2( 4) = 8, a large positive number, as expected

6 copyrigt Dr Carol JV Fiser Burns ttp://wwwonematematicalcatorg 187 EXERCISE 3 3 Grap te function f(x) = x 2 2 Wat do you expect for te slope of te tangent line wen x = 0? Wen x is a small positive number? Wen x is a large negative number? 3 Using te it formula, calculate te slope of te tangent line at a typical point (x, f(x)) 4 Wat is te slope of te tangent line at (x, f(x))? Does tis agree wit your expectations? caracterizing a two-sided it by using one-sided its Suppose a function g is defined bot to te left and to te rigt of c In order for te two-sided it x c g(x) to exist, te function values g(x) must approac te same number as x approaces c coming in from bot sides Tat is, te two-sided it x c g(x) exists exactly wen bot one-sided its g(x) and g(x) x c + x c exist, and ave te same value Tis observation is used in te next examples

7 188 copyrigt Dr Carol JV Fiser Burns ttp://wwwonematematicalcatorg EXAMPLE A function wic does not ave a tangent line at a point Consider te function f defined piecewise as follows: Te grap of f is sown below { 2x 1 wen x 2 f(x) = 1 3 x wen x < 2 First consider a point (x, f(x)) wen x > 2 In tis case, to te immediate left and rigt of te point (x, f(x)) te function f looks like ( Wy?) Tus, we find tat f(x) = 2x 1 ( ) 2(x + ) 1 (2x 1) = 0 = ( You fill in te details) = 2 Similarly, if x < 2, te slopes of tangent lines are all 1 3 ( Be sure to ceck tis yourself) Te interesting situation occurs wen x = 2; let us now investigate te it f(2 + ) f(2) Remember tat tis it is, in general, a 2-sided object Since te function f being investigated IS defined bot to te rigt ( > 0) and left ( < 0) of 2, we must see wat appens as approaces 0 from te rigt-and side and te left-and side Wenever > 0 ( approaces 0 from te rigt-and side), we ave 2 + > 2, so tat f(2 + ) f(2) (2(2 + ) 1) 3 = = 2 and so f(2 + ) f(2) = 2 0 +

8 copyrigt Dr Carol JV Fiser Burns ttp://wwwonematematicalcatorg 189 Wenever < 0 (so tat approaces 0 from te left-and side), we ave 2 + < 2, so tat f(2 + ) f(2) = ( (2 + ) + 3 ) 3 = 1 3 and so f(2 + ) f(2) = Since te rigt and left and its do not agree, te two-sided it f(2 + ) f(2) does not exist Tat is, tere is no tangent line to f at x = 2 expected! Tis result was, of course, EXERCISE 4 Consider te function f, wit grap sown below 1 Give a piecewise description for tis function f Now, attempt to find te tangent line at te point (1, 3), as follows: f(1 + ) f(1) 2 Find 0 + f(1 + ) f(1) 3 Find 0 f(1+) f(1) 4 Does exist? Wy or wy not?

9 190 copyrigt Dr Carol JV Fiser Burns ttp://wwwonematematicalcatorg EXERCISE 5 Wen is a number near zero, x + is a number near x So, in evaluating te it, we require tat f be defined on some interval containing x Tis interval can be of any of tese forms: 1 If f is defined on an interval (a, b) containing x, ten is te it a genuine two-sided it? Wy or wy not? 2 If f is only defined on an interval of te form [x, b), ten is te it f(x+) f(x) a genuine two-sided it? If not, wat type of it is it? 3 If f is only defined on an interval of te form (a, x], ten is te it f(x+) f(x) a genuine two-sided it? If not, wat type of it is it? identifying lines ALGEBRA REVIEW point-slope form for lines Two pieces of (non-contradictory, non-overlapping) information uniquely determine a line Te most common information given to identify a line is: two distinct (different) points on te line; or te slope of te line, and a point on te line Suppose tat te slope of a line is known, call it m; and a point on te line is known, call it (x 1, y 1 ) Now, let (x, y) denote any oter point on te uniquely identified line (so x x 1 ) Using te points (x 1, y 1 ) and (x, y) to compute te (known) slope: y y 1 x x 1 = m y y 1 = m(x x 1 ) point-slope form of a line EXAMPLE using point-slope form Tus, any point (x, y) lying on te line wit slope m troug (x 1, y 1 ) makes te equation y y 1 = m(x x 1 ) true; and any point tat makes te equation true lies on te line Tat is, te equation of a line tat as slope m and passes troug te point (x 1, y 1 ) is given by y y 1 = m(x x 1 ) ; tis is called te point-slope form of a line Problem: Find te equation of te line tat as slope 2, and passes troug te point ( 1, 3) Solution: Te information is ideally suited to point-slope form: y 3 = 2(x ( 1)) y = 3 + 2(x + 1) y = 2x + 5

10 copyrigt Dr Carol JV Fiser Burns ttp://wwwonematematicalcatorg 191 Problem: Find te equation of te line tat passes troug te points (5, 2) and ( 1, 3) Solution: First, find te slope of te line: m = 3 ( 2) ( 1) 5 = 5 6 = 5 6 Ten, use eiter point, te known slope, and point-slope form Using te point (5, 2), te equation is y ( 2) = 5 (x 5) 6 Using te point ( 1, 3), te equation is y 3 = 5 (x ( 1)) 6 EXERCISE 6 Verify tat te two equations obtained above are equivalent; tat is, tey describe precisely te same line Tat is, sow tat y ( 2) = 5 6 (x 5) y 3 = 5 (x ( 1)) 6 One way to do tis is to put bot equations into te same form; say, y = mx + b form, or ax + by + c = 0 form Once tey re in te same form, tey are easy to compare QUICK QUIZ sample questions 1 Use a it to compute te slope of te tangent line to te grap of f(x) = x at x = 2 Be sure to write complete matematical sentences 2 In te expression, wat is te dummy variable? Rewrite te it using a different dummy variable (you coose) 3 In te expression, wat does x represent? 4 In te it, wat does f(x+) f(x) represent? 5 Let f : [0, 3] R be defined by f(x) = x 2 Grap f Does f(0 + ) f(0) exist? If so, wat is it? KEYWORDS for tis section Tangent lines, finding te slopes of tangent lines, secant lines, difference quotient, slope of te tangent line to te grap of a function f at te point (x, f(x)), caracterizing a two-sided it by using one-sided its

11 192 copyrigt Dr Carol JV Fiser Burns ttp://wwwonematematicalcatorg END-OF-SECTION EXERCISES Classify eac entry below as an expression (EXP) or a sentence (SEN) For any sentence, state weter it is TRUE, FALSE, or CONDITIONAL 1 2 g(x + x) g(x) x 0 3 = m g(x + x) g(x) 4 = m x 0 5 Te slope of te tangent line to te grap of f(x) = x 2 at te point (x, x 2 ) equals 2x 6 Te slope of te tangent line to te grap of g(x) = 5 at te point (x, 5) equals 0 For te remaining problems, define a function g by g() :=, were f is a function of one variable, wit x D(f) 7 Find g(1) and g( x) 8 Rewrite te it in terms of te function g 9 Wen is a number in te domain of g? Answer using a complete matematical sentence 10 Wat does te number g() tell us? 11 Wat does te number 0 g() tell us, wen it exists? 12 Write down te ɛ-δ definition of te sentence g() = m 0 Be sure to write a complete matematical sentence