. Compute the following limits.
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1 Today: Tangent Lines and te Derivative at a Point Warmup:. Let f(x) =x. Compute te following limits. f( + ) f() (a) lim f( +) f( ) (b) lim. Let g(x) = x. Compute te following limits. g(3 + ) g(3) (a) lim g( 5+) g( 5) (b) lim Answers: (a), (b) 8; (a) 9,(b) 5
2 Recall: Average velocity 5 Te average velocity from t = t to t = t is cange in distance avg velocity = cange in time = slope of secant line Plot position f versus time t: y = f(t) f(a) f(a+) a a+ Pick two points on te curve (a, f(a)) and (b, f(b)). Rewrite b = a +. Slope of te line connecting tem: f(a + ) avg velocity = m = f(a) Te smaller is, te more useful m is! di erence quotient
3 Te di erence quotient (average velocity) of te curve y = f(x) at x = a is f(a + ) avg velocity = f(a). Te slope (instantaneous velocity) of te curve y = f(x) at te point (a, f(a)), calledtederivative of f(x) at x = a, iste number f 0 f(a + ) f(a) (a) =lim (provided te limit exists). Te tangent line ` to te curve at (a, f(a)) is te line troug (a, f(a)) wit tis slope: ` : y f(a) =m(x a), were m = f 0 (a). (Recall point-slope form: y y 0 = m(x x 0 ).)
4 Te slope (instantaneous velocity) of te curve y = f(x) at te point (a, f(a)), calledtederivative of f(x) at x = a, iste number f 0 f(a + ) f(a) (a) =lim (provided te limit exists). Recall te warmup:. Let f(x) =x. Compute te following limits. (a) lim!0 f( + ) f() (b) lim!0 f( +) f( ) = slope at x = = 4 slope at x = blue line: y = (x ) red line: y 4= 4(x + ) Te slope (instantaneous velocity) of te curve y = f(x) at te point (a, f(a)), calledtederivative of f(x) at x = a, iste number f 0 f(a + ) f(a) (a) =lim (provided te limit exists). Recall te warmup:. Let g(x) = x. Compute te following limits. (a) lim!0 g(3 + ) g(3) (b) lim!0 g( 5+) g( 5) = /9 slope at x =3 = /5 slope at x = blue line: y 3 = 9 (x 3) red line: y + 5 = (x + 5) 5 -
5 You try: For eac of te following examples... (a) Compute f 0 (a) using te formula f 0 f(a + ) f(a) (a) =lim. (b) Compute te equation for te tangent line to (a, f(a)) using point-slope form. (c) Sketc y = f(x) near x = a and te line you computed in part (b) on te same set of axes to ceck tat your answers make sense.. f(x) =x 3 at a =0, a =,anda =.. f(x) = p x at a =and a =4. 3. f(x) =sin(x) at a =0, a = /4 and a = /. [For 3, recall sin( + lim!0 sin( )/ =.] )=sin( ) cos( ) + cos( )sin( ) and
6 Wen can we take derivatives? Not all functions ave derivatives at all places. Before calculating f 0 (a), firstask... () Is f(x) defined at x = a? For example, even if it looks like you could draw a tangent line, if tere s a ole, f 0 (a) does not exist a NO DERIVATIVE! (It s tempting to say f 0 (a) exists ere in part because f(x) as a continuous extension at a.) Wen can we take derivatives? Not all functions ave derivatives at all places. Before calculating f 0 (a), firstask... () Is f(x) continuous at x = a? For example, even if it looks like you could draw a tangent line, if tere s a jump, f 0 (a) does not exist! a NO DERIVATIVE! (Try drawing just one line tat is tangent to tat isolated point. It s tempting to say f 0 (a) exists ere in part because f(x) as a removable discontinuity at a.)
7 Wen can we take derivatives? Not all functions ave derivatives at all places. Before calculating f 0 (a), firstask... () Is f(x) continuous at x = a? Again, even if te slope looks te same from te left and from te rigt, if tere s a discontinuity, f 0 (a) does not exist! a NO DERIVATIVE! Wen can we take derivatives? Not all functions ave derivatives at all places. Before calculating f 0 (a), firstask... (3) Is tere a corner at x = a? Next we ll explore ow to find tese algebraically, but if tere s asarpcorneratx = a, ten f 0 (a) does not exist! a NO DERIVATIVE! (Try drawing just one line tat is tangent to tat corner)
8 Wat s wrong wit corners? You try: Let f(x) = ( x x<, x + x. (a) Verify tat f(x) is continuous at x =. (Compute lim x! f(x), lim x! + f(x), andf(), and compare.) (b) Sketc a grap of f(x). (c) Estimate (ok to use a calculator), and ten calculate te rigt sided derivative. (d) Estimate (ok to use a calculator), and ten calculate te left sided derivative. (e) Compare your answers to (c) and (d), and explain wy f( + ) f() lim does not exist. Explain wy f 0 () does not exist. Estimate te rigt-sided derivative: f() = f( + ) f( + ) f() / /0 f(+) f() Compute te rigt-sided derivative: f( + ) f() lim =!0 + Estimate te left-sided derivative: f() = f( + ) f( + ) f() / /0 Compute te rigt-sided derivative: f( + ) f() lim = f(+) f()
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