Overview We alread saw an algebraic wa of thinking about a derivative. Geometric: zooming into a line Analtic: continuit and rational functions Computational: approimations with computers
3. The geometric view: zooming into the graph of a function Zooming In For a smooth function, if we zoom in at a point, we see a line: 2 2.5.25.25 In this eample, the slope of our zoomed-in line looks to be about: 2
3. The geometric view: zooming into the graph of a function Recall The secant line to the curve = f () through points R and Q is a line that passes through R and Q. We call the slope of the secant line the average rate of change of f () from R to Q. R Q P secant line tangent line Definition The straight line that we see when we zoom into the graph of a smooth function at some point P is called the tangent line at P. The slope of the tangent line is the instantaneous rate of change (derivative)
3. The geometric view: zooming into the graph of a function On the graph below, draw the secant line to the curve through points P and Q. On the graph below, draw the tangent line to the curve at point P. P Q P
3. The geometric view: zooming into the graph of a function Derivatives of Familiar Functions The derivative of the function f () = A is: (a) 0 (b) (c) A (d) (e) A 2 The derivative of the function f () = A is: (a) 0 (b) (c) A (d) (e) A 3 The derivative of the function f () = A + is: (a) 0 (b) (c) A (d) (e) A
3. The geometric view: zooming into the graph of a function km from home 6 3 = s(t) 0 8:00 8:07 8:30 It takes me half an hour to bike 6 km. So, 2 kph represents the: A. secant line to = s(t) from t = 8 : 00 to t = 8 : 30 t time B. slope of the secant line to = s(t) from t = 8 : 00 to t = 8 : 30 C. tangent line to = s(t) at t = 8 : 30t = 8 : 25 D. slope of the tangent line to = s(t) at t = 8 : 30
3. The geometric view: zooming into the graph of a function km from home 6 3 = s(t) 0 8:00 8:07 8:30 t time When I pedal up Crescent Rd at 8:25, the speedometer on m bike tells me I m going 5kph. This corresponds to the: A. secant line to = s(t) from t = 8 : 00 to t = 8 : 25 B. slope of the secant line to = s(t) from t = 8 : 00 to t = 8 : 25 C. tangent line to = s(t) at t = 8 : 25 D. slope of the tangent line to = s(t) att = 8 : 25
Photo credit: Piaba 3. The geometric view: zooming into the graph of a function Zooming in on functions that aren t smooth For a function with a cusp or a discontinuit, even though we zoom in ver closel, we don t see simpl a single straight line. Cusp: 2.5.25 Discontinuit: 2.5.25
3. The geometric view: zooming into the graph of a function Sketch a derivative Sketch the derivative of the function shown. = f ()
3. The geometric view: zooming into the graph of a function Sketch a derivative Sketch the derivative of the function shown. = f () = f () 0 4 2 4 0
3. The geometric view: zooming into the graph of a function Sketch a derivative Sketch the derivative of the function shown. = f ()
3. The geometric view: zooming into the graph of a function Sketch a derivative Sketch the derivative of the function shown. = f () = f ()
3. The geometric view: zooming into the graph of a function Sketch a derivative Sketch the derivative of the function shown. = f ()
3. The geometric view: zooming into the graph of a function Sketch a derivative Sketch the derivative of the function shown. = f () = f ()
3. The geometric view: zooming into the graph of a function Sketch a derivative Sketch the approimate behaviour of the function, then use our sketch of the function to sketch its derivative. f () = 5 2 0 + 2
3. The geometric view: zooming into the graph of a function Kinesin Kinesin: link Kinesin transports vesicles along microtubules, which have a plus and minus end Kinesin onl travels towards the plus end Kinesin can hop on and off microtubules + + More eplanation of Kinesin s walking: link
3. The geometric view: zooming into the graph of a function Kinesin Below is a sketch of the position of a kinesin molecule over time, as it travels around a network of microtubules of fied position. a b c t Sketch the velocit of the kinesin molecule over the same time period. 2 Describe what happened to the kinesin molecule.
3. The geometric view: zooming into the graph of a function Kinesin Below is a sketch of the position of a kinesin molecule over time, as it travels around a network of microtubules of fied position. velocit a b c t (0, a) The molecule travels along a microtubule at a velocit of in the positive direction. (a, b) Then, it falls off or gets stuck. (b, c) Then it travels on along a mictorubule with the opposite orientation, so it moves in the negative direction. t > c Finall, it hops onto another microtubule (possibl the same as the first) and travels in the positive direction.
3.2 The analtic view: calculating the derivative Overview Eplain the definition of a continuous function. Identif functions with various tpes of discontinuities. Evaluate simple limits of rational functions. Calculate the derivative of a simple function using the definition of the derivative
3.2 The analtic view: calculating the derivative Intuitive Continuit Intuitivel, we think of a point on a function as continuous if it s connected to the points around it. Jump Discontinuit
3.2 The analtic view: calculating the derivative Intuitive Continuit Intuitivel, we think of a point on a function as continuous if it s connected to the points around it. Removable Discontinuit
3.2 The analtic view: calculating the derivative Intuitive Continuit Intuitivel, we think of a point on a function as continuous if it s connected to the points around it. Blow-Up (Infinite) Discontinuit If a graph has a discontinuit at a point, it has no tangent line at that point (and so no derivative at that point).
3.2 The analtic view: calculating the derivative Intuition Failing { sin ( ) f () = if 0 0 if = 0
3.2 The analtic view: calculating the derivative Intuition Failing { sin ( ) f () = if 0 0 if = 0
3.2 The analtic view: calculating the derivative Intuition Failing f () = { 3 if is rational 3 if is irrational
3.2 The analtic view: calculating the derivative A More Rigorous Definition Definition A function f () is continuous at a point a in its domain if lim f () = f (a) a f (a) must eist As approaches a, there are no surprises. Jump: Limit doesn t eist (left and right don t match)
3.2 The analtic view: calculating the derivative Continuit Proofs Decide whether f () = 2 4 2 Justif our result. is continuous or discontinuous at = 2. It is discontinuous, because 2 is not in the domain of f (). That is, f (2) does not eist. Eample : 2 4 if 2 For which value(s) of a is f () = 2 a if = 2 Prove our result. continuous at = 2?
3.2 The analtic view: calculating the derivative Rational Functions with Holes Eample 2: Let f () = ( + )( )( 2) ( )( 2) Note the domain of f () does not include = or = 2. Evaluate the following: (a) lim 0 f () (b) lim f () (c) lim 2 f () (d) lim 3 f () 2
3.2 The analtic view: calculating the derivative Using limits to evaluate derivatives Evaluate the derivative of f () = using the definition of the derivative, f () = lim h 0 f ( + h) f () h f f ( + h) f () + h () = lim = lim h 0 h h 0 h ( ) + h + h + = lim h 0 h + h + = lim h 0 ( + h) h( + h + ) = lim h 0 = lim h 0 ( + h + ) = = + 0 + 2 h h( + h + )
3.2 The analtic view: calculating the derivative Using limits to evaluate derivatives Eample 3: Find the derivatives of the following functions using the definition of a derivative: (a) f () = (b) g() = f () = lim h 0 f ( + h) f () h
3.2 The analtic view: calculating the derivative Concept Check True or False: If f () is not defined at =, then f () does not eist. lim True or False: If lim f () eists, then it s equal to f (). True or False: If f () is continuous at =, then f () eists. True or False: If f () eists, then f () is continuous at =.
3.2 The analtic view: calculating the derivative Using Graphs Eample 4: Approimate sin lim 0 using the graph below.
3.3 The computational view: software to the rescue! Overview. Use software to numericall compute an approimation to the derivative. 2 Eplain that the approimation replaces a (true) tangent line with an (approimating) secant line. 3 Eplain using words how the derivative shape is connected with the shape of the original function.
3.3 The computational view: software to the rescue! Idea: Approimate Tangent Line with Secant Line Derivative (slope of tangent line) f ( + h) f () lim h 0 h Approimate derivative Slope of secant line f ( + h) f () h
3.3 The computational view: software to the rescue! Approimate Derivative f () = 2 5 2 5 + 5 f f ( + h) f () () = lim = 320 h 0 h 33 2 0.29385... h f( + h) f() h 0.93939... 0. 0.35228... 0.0 0.29932... 0.00 0.29439... 0.000 0.29390... 0.0000 0.29385... spreadsheet link
3.3 The computational view: software to the rescue! Approimate Derivative f () = 2 5 2 5 + 5 f f ( + h0.) f () () h0. Using ver small values of h can get us an approimation of the derivative at a particular point. We can also use a spreadsheet to guess the derivative at a number of different points, using tin secant lines..25 0. 0.
3.3 The computational view: software to the rescue! Approimate derivative at man points Interval: [, 4] Small h: sa h = 0. f () = 2 5 2 5 + 5 2 3 4 spreadsheet link
3.3 The computational view: software to the rescue! Predator-Pre Interactions: Holling Predator Response predation rate P() = 3 3 + 3 pre densit Tpe III: I can hardl find the pre when the pre densit is low, but I also get satiated at high pre densit. (a) Label the asmptote on the -ais. (b) Label the half-ma activation level on the -ais. (c) At what pre densit is the predation rate changing the fastest? That is, at which densit is each additional pre animal responsible for the