Unit #3 : Differentiability, Computing Derivatives Goals: Determine when a function is differentiable at a point Relate the derivative graph to the the graph of an original function Compute derivative functions of powers, exponentials, logarithms, and trig functions. Compute derivatives using the product rule, quotient rule, and chain rule for derivatives. Textbook reading for Unit #3 : Study Sections 1.5,2.3,2.4,2.6 and 3.1 3.6.
Unit 3-2 In the previous unit we introduced the definition of the derivative. In this unit we will use and compute the derivative more efficiently. As a lead in, though, let us review how we arrived at the derivative concept. Interpretations of Secants vs Derivatives From Section 2.4 The slope of a secant line gives the average rate of change of f(x) over some interval x. the average velocity over an interval, if f(t) represents position. the average acceleration over an interval, if f(t) represents velocity. Give the units of the slope of a secant line.
Unit 3-3 The derivative gives the limit of the average slope as the interval x approaches zero. a formula for slopes for the tangent lines to f(x). the instantaneous rate of change of f(x). the velocity, if f(t) represents position. the acceleration, if f(t) represents velocity. Give the units of the derivative.
Unit 3-4 Differentiability From Section 2.6 Recall the definition of the derivative. f (x) = df dx = lim f x 0 x = lim f(x+h) f(x) h 0 h A function f is differentiable at a given point a if it has a derivative at a, or the limit above exists. There is also a graphical interpretation differentiability: if the graph has a unique and finite slope at a point. Since the slope in question is automatically the slope of the tangent line, we could also say that f is differentiable at a if its graph has a (non-vertical) tangent at (a,f(a)). Forfunctions oftheform y = f(x), wedo notconsiderpoints withvertical tangent lines to have a real-valued derivative, because a vertical line does not have a finite slope.
Unit 3-5 Here are the ways in which a function can fail to be differentiable at a point a: 1. The function is not continuous at a. 2. The function has a corner (or a cusp) at a. 3. The function has a vertical tangent at (a,f(a)). Sketch an example graph of each possible case.
Unit 3-6 Investigate the limits, continuity and differentiability of f(x) = x at x = 0 graphically.
Use the definition of the derivative to confirm your graphical analysis. Unit 3-7
Unit 3-8 We have seen at a point that a function can have, or fail to have, the following descriptors: continuous; limit exists; is differentiable. Put these properties in decreasing order of stringency, and sketch relevant illustrations.
Unit 3-9 Differentiability is Common You will notice that, despite our concern about some functions not being differentiable, most of our standard functions (polynomials, rationals, exponentials, logarithms, roots) are differentiable at most points. Therefore we should investigate what all these possible derivative/slope values could tell us. Interpreting the Derivative From Section 2.4 Where f (x) > 0, or the derivative is positive, f(x) is increasing. Where f (x) < 0, or the derivative is negative, f(x) is decreasing. Where f (x) = 0, or the tangent line to the graph is horizontal, f(x) has a critical point.
Unit 3-10 A B C Question 1 f (x) > 0 on the intervals 1. A C, E G 2. C E 3. B D, F G 4. A B, D F D E F G
Unit 3-11 A B C D E F G Question 2f (x)takesonitslargest negative value at 1. A 2. B 3. C 4. D 5. G
Unit 3-12 Graphs, and Graphs of their Derivatives From Section 2.3 Example: Consider the same graph again, and the graph of its derivative. Identify important features that associate the two.
Unit 3-13 A B C D E F G A B C D E F G
Unit 3-14 Question: Consider the graph of f(x) shown: 2 1 1 0 1 2 1 Which of the following graphs is the graph of the derivative of f(x)?
Unit 3-15 2 2 1 1 1 0 1 2 1 0 1 2 2 1 1 1 1 0 1 2 1 2 1 A 2 1 B 1 0 1 2 1 0 1 2 1 1 C D
Unit 3-16 Note: The standard formulas for derivatives are covered in the Grade 12 Ontario curriculum. While they will be reviewed here, students who are not familiar with them should begin both textbook reading and the assignment problems for this unit as soon as possible. Computing Derivatives From Sections 3.1-3.6 Beyond the graphical interpretation of derivatives, there are all the algebraic rules. All of these rules are based on the definition of the derivative, f (x) = df dx = lim f x 0 x = lim f(x+h) f(x) h 0 h However, by finding common patterns in the derivatives of certain families of functions, we can compute derivatives much more quickly than by using the definition.
Unit 3-17 Sums, Powers, and Differences d Constant Functions: dx k = 0 d Power rule: dx xp = px p 1 ( ) ( ) d d d Sums : dx f(x)+g(x) = dx f(x) + ( ) ( dx g(x) ) d d d Differences: dx f(x) g(x) = dx f(x) ( ) dx g(x) d d Constant Multiplier: dx kf(x) = k dx f(x), k a constant
Unit 3-18 Example: Evaluate the following derivatives: d ( x 4 +3x 2) dx d dx ( 2.6 x πx 3 +4 )
Unit 3-19 Question: The derivative of 3x 2 1 x 2 is 1. 6x 3 +2 1 x 3 2. 6x+2 1 x 3 3. 6x 2 1 x 3 4. x 3 +2 1 x
Unit 3-20 Exponentials and Logs e as a base: Other bases: Natural Log: Other Logs: d dx ex = e x d dx ax = a x (ln(a)) d dx ln(x) = 1 x d dx log a(x) = 1 1 xln(a)
Unit 3-21 Example: Evaluate the following derivatives: d ( 4 10 x +10 x 4) dx d dx (ex +log 10 (x)) (Exponential and log derivatives are relatively straightforward, until we mix in the product, quotient, and chain rules.)
Unit 3-22 Product and Quotient Rules Products: d dx f(x) g(x) = f (x)g(x)+f(x)g (x) Quotients: d dx f(x) g(x) = f (x)g(x) f(x)g (x) (g(x)) 2 Example: Evaluate the following derivatives: d ( 4x 2 e x) dx
Unit 3-23 d dx (xln(x)) d dx (5 x2 ln(x) )
Unit 3-24 Question: The derivative of 10x x is 3 10 x 1. ln(10) x 3 +10 x ( 3x 4 ) 2. 10x ln(10)x 3 10 x (3x 2 ) x 6 3. 10x 1 ln(10) x3 10 x (3x 2 ) x 6 4. ln(10)10 x x 3 +10 x ( 3x 4 )
Unit 3-25 Chain Rule Nested Functions: d dx [f(g(x))] = f (g(x)) g (x) Liebnitz form d dx df dg f(g(x)) = dgdx
Unit 3-26 Example: Evaluate the following derivatives: d dx ex2
d dx ln(x4 ) Unit 3-27
Unit 3-28 d dx ( ) 1 1+x 3
Unit 3-29 d dx ( x 4 +10 3x)
Unit 3-30 Question: The derivative of e x is 1. 1 2 e 1 x 2. e ( x x ) 3. 1 ( ) 2 e x 1 x 4. 1 2 e x ( x )
Unit 3-31 Trigonometric Functions From Section 1.5 In our earlier discussion of functions, we skipped over the trigonometric functions. We return to them now to discuss both their properties and their derivative rules. The trigonometric functions are usually defined for students first using triangles (recall the mnemonic device, SOHCAHTOA ).
Unit 3-32 Use the 45/45 and 60/30 triangles to compute the sine and cosine of these common angles.
Unit 3-33 Extending Trigonometric Domains One difficulty with limiting ourselves to the triangle ratio definition of the trig functions is that the possible angles are limited to the range θ [0, π 2 ] radians or θ [0,90] degrees. To remove this limitation, mathematicians extended the definition of the trigonometric functions to a wider domain via the unit circle. θ
Unit 3-34 How does the circle definition lead to the trigonometric identity sin 2 (θ) + cos 2 (θ) = 1?
Unit 3-35 Show how the circle and triangle definitions define the same values in the first quadrant of the unit circle. It is useful to understand both definitions of trig functions (circle and triangle) as sometimes one is more helpful than the other for a particular task.
Unit 3-36 Sine and Cosine as Oscillating Functions Despite the geometric source of the trigonometric functions, they are used more commonly in biology and many other sciences as because their periodicity and oscillatory shapes. For many cyclic behaviours in nature, trigonometric functions are a natural first choice for modeling.
Unit 3-37 Question The graph of y = 10 + 4cos(x) is shown in which of the following diagrams? 14 12 10 8 6 4 2 2 4 14 12 10 8 6 4 2 6 A B 12 12 10 8 6 4 2 10 8 6 4 2 C D Show the amplitude and the average on the correct graph.
Unit 3-38 Period and Phase How can you find the period of the function cos(ax)?
Unit 3-39 How can you reliably determine where the function cos(ax + B) starts on the graph? (For a cosine graph, where the start represents a maximum, the starting time or x value is sometimes called the phase of the function.)
Unit 3-40 Consider the graph of the function y = 5 + 8cos(π(x 1)). following properties of the function: amplitude What are the period average phase
Unit 3-41 Sketch the graph on the axes below. Include at least one full period of the function.
Unit 3-42 More complicated amplitudes In the form y = A + Bcos(Cx + D), the B factor sets the amplitude. In many interesting cases, however, that amplitude need not be constant. Sketch the graph of y = 5, and the graph of y = 5cos(x) on the axes below.
Unit 3-43 Sketch the graph of y = x, and the graph of y = xcos(πx) on the axes below. Use only x 0
Unit 3-44 Use your intuition to sketch the graph of y = e x cos(πx) on the axes below.
Unit 3-45 Derivatives of Trigonometric Functions From Section 3.5 Having covered the graphs and properties of trigonometric functions, we can now review the derivative formulae for those same functions. The derivation of the formulas for the derivatives of sin and cos are an interesting study in both limits and trigonometric identities. For those who are interested, manysuch derivations can be found on the web 1. However, it is in some ways more useful to derive the formula in a graphical manner. 1 For example, http://www.math.com/tables/derivatives/more/trig.htm#sin
Unit 3-46 Below is a graph of sin(x). Use the graph to sketch the graph of its derivative. 1 3 π /2 π π/2 0 π/2 π 3 π/2 1 1 3 π /2 π π/2 0 π/2 π 3 π/2 1
Unit 3-47 From this sketch, we have evidence (though not a proof) that Theorem d dx sinx =
Unit 3-48 Most students will also be familiar with the other derivative rules for trig functions: d cos(x) = sin(x) dx d dx tan(x) = sec2 (x) d sec(x) = sec(x)tan(x) dx d dx csc(x) = csc(x)cot(x) d dx cot(x) = csc2 (x)
Unit 3-49 Prove the secant derivative rule, using the definition sec(x) = 1 cos(x) other derivative rules. and the
Unit 3-50 Question: Find the derivative of 4+6cos(πx 2 +1) 1. 4 6sin(πx 2 +1) (2πx) 2. 6cos(πx 2 +1) (2πx) 3. 6sin(πx 2 +1) (2πx) 4. 6sin(πx 2 +1) (πx 2 +1) 5. 6 sin(2πx)
Unit 3-51 Inverse Trig Functions From Section 1.5, 3.6 In addition to the 6 trig functions just seen, there are 6 inverse functions as well, though the inverses of sine, cosine, and tangent are the most commonly used. Sketch the graph of sin(x) on the axes below On the same axes, sketch the graph of arcsin(x), or sin 1 x, or the inverse of sin(x).
Unit 3-52 What is the domain of arcsin(x)? What is the range of arcsin(x)?
Unit 3-53 Inthenextfewquestionsyouwillobtaintheformulaforthederivativeofarcsinx. Simplify sin(arcsin x) Differentiate both sides of this equation, using the chain rule on the left. You should end up with an equation involving d dx arcsinx.
Unit 3-54 Solve for d dx formula arcsinx, and simplify the resulting expression by means of the which is valid if θ [ π 2, π 2 ]. cosθ = 1 sin 2 θ, d dx arcsinx =