Secants vs. Derivatives - Unit #3 : Goals: Differentiability, Computing Derivatives, Trig Review Determine when a function is ifferentiable at a point Relate the erivative graph to the the graph of an original function Compute erivative functions of powers, exponentials, logarithms, an trig functions. Compute erivatives using the prouct rule, quotient rule, an chain rule for erivatives. Review trigonometric an inverse trigonometric functions. Textbook reaing for Unit #3 : Stuy Sections.5,.3,.4,.6 an 3. 3.6. In the previous unit we introuce the efinition of the erivative. In this unit we will use an compute the erivative more efficiently. As a lea in, though, let us review how we arrive at the erivative concept. Interpretations of Secants vs Derivatives From Section.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. Secants vs. Derivatives - The erivative 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 erivative. Differentiability From Section.6 Recall the efinition of the erivative. f (x) = x f = f x = lim x 0 f x = lim h 0 f(x + h) f(x) h Differentiability - A function f is ifferentiable at a given point a if it has a erivative at a, or the limit above exists. There is also a graphical interpretation ifferentiability: if the graph has a unique an finite slope at a point. Since the slope in question is automatically the slope of the tangent line, we coul also say that f is ifferentiable at a if its graph has a (non-vertical) tangent at (a, f(a)). For functions of the form y = f(x), we o not consier points with vertical tangent lines to have a real-value erivative, because a vertical line oes not have a finite slope.
Differentiability - Here are the ways in which a function can fail to be ifferentiable at a point a:. The function is not continuous at a.. 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. Differentiability - 3 Investigate the limits, continuity an ifferentiability of f(x) = x at x = 0 graphically. Use the efinition of the erivative to confirm your graphical analysis. Differentiability - 4 Differentiability - 5 We have seen at a point that a function can have, or fail to have, the following escriptors: continuous; limit exists; is ifferentiable. Put these properties in ecreasing orer of stringency, an sketch relevant illustrations.
Differentiability - 6 Reminer: Differentiability is Common You will notice that, espite our concern about some functions not being ifferentiable, most of our stanar functions (polynomials, rationals, exponentials, logarithms, roots) are ifferentiable at most points. Therefore we shoul investigate what all these possible erivative/slope values coul tell us. Interpreting the Derivative From Section.4 Interpreting the Derivative - Where f (x) > 0, or the erivative is positive, f(x) is increasing. Where f (x) < 0, or the erivative is negative, f(x) is ecreasing. Where f (x) = 0, or the tangent line to the graph is horizontal, f(x) has a critical point. Interpreting the Derivative - Interpreting the Derivative - 3 Example: A B Consier the graph of f(x) shown below. A B C On what intervals is f (x) > 0? C Where oes f (x) takes on its largest negative value? D E F G D E F G
Graphs of the Derivative - Graphs of the Derivative - Graphs, an Graphs of their Derivatives Example: Consier the same graph again, an the graph of its erivative. Ientify important features that associate the two. B A Question: Consier the graph of f(x) shown: C 0 D E F G A B C D E F G Which of the following graphs is the graph of the erivative of f(x)? Graphs of the Derivative - 3 Computing Derivatives - Basic Formulas - Computing Derivatives 0 0 A 0 0 B 0 Note: The stanar formulas for erivatives are covere in the Grae Ontario curriculum. While they will be reviewe here, stuents who are not familiar with them shoul begin both textbook reaing an the assignment problems for this unit as soon as possible. From Sections 3.-3.6 Beyon the graphical interpretation of erivatives, there are all the algebraic rules. All of these rules are base on the efinition of the erivative, f (x) = x f = f x = lim f x 0 x = lim f(x + h) f(x) h 0 h However, by fining common patterns in the erivatives of certain families of functions, we can compute erivatives much more quickly than by using the efinition. C D
Computing Derivatives - Basic Formulas - Computing Derivatives - Basic Formulas - 3 Sums, Powers, an Differences Constant Functions: x k = 0 Power rule: x xp = px p ( ) ( ) Sums : f(x) + g(x) = x x f(x) + ( ) ( x g(x) ) Differences: f(x) g(x) = x x f(x) ( ) x g(x) Constant Multiplier: x [kf(x)] = k x f(x), so long as k is a constant Example: Evaluate the following erivatives: ( x 4 + 3x ) x (.6 x πx 3 + 4 ) x Computing Derivatives - Basic Formulas - 4 Computing Derivatives - Basic Formulas - 5 Question: The erivative of 3x x is. 6x 3 + x 3. 6x + x 3 3. 6x x 3 4. x 3 + x Exponentials an Logs e as a base: Other bases: Natural Log: Other Logs: x ex = e x x ax = a x (ln(a)) x ln(x) = x x log a(x) = xln(a)
Computing Derivatives - Basic Formulas - 6 Computing Derivatives - Prouct an Quotient Rules - Example: Evaluate the following erivatives: ( 4 0 x + 0 x 4) x Prouct an Quotient Rules Proucts: x f(x) g(x) = f (x)g(x) + f(x)g (x) x (ex + log 0 (x)) Example: ( 4x e x) x Quotients: f(x) x g(x) = f (x)g(x) f(x)g (x) (g(x)) Evaluate the following erivatives: (Exponential an log erivatives are relatively straightforwar, until we mix in the prouct, quotient, an chain rules.) Computing Derivatives - Prouct an Quotient Rules - Computing Derivatives - Prouct an Quotient Rules - 3 (x ln(x)) x ) (5 x x ln(x) Question: The erivative of 0x x 3 0 x. ln(0) x 3 + 0 x ( 3x 4 ). 0x ln(0)x 3 0 x (3x ) x 6 3. 0x ln(0) x3 0 x (3x ) x 6 4. ln(0)0 x x 3 + 0 x ( 3x 4 ) is
Computing Derivatives - Chain Rule - Computing Derivatives - Chain Rule - Chain Rule Neste Functions: x [f(g(x))] = f (g(x)) g (x) Example: x ex Evaluate the following erivatives: Liebnitz form f g f(g(x)) = x g x x ln(x4 ) Computing Derivatives - Chain Rule - 3 ( ) x + x 3 Computing Derivatives - Chain Rule - 4
x Computing Derivatives - Chain Rule - 5 Computing Derivatives - Chain Rule - 6 ( x 4 + 0 3x) Question:. e x. e ( x x ) 3. ( ) e x x 4. e x ( x ) The erivative of e x is Trigonometry Review Trigonometry Review - From Section.5 In our earlier iscussion of functions, we skippe over the trigonometric functions. We return to them now to iscuss both their properties an their erivative rules. The trigonometric functions are usually efine for stuents first using triangles (recall the mnemonic evice, SOHCAHTOA ). Trigonometry Review - Use the 45/45 an 60/30 triangles to compute the sine an cosine of these common angles.
Trigonometry Review - 3 Extening Trigonometric Domains One ifficulty with limiting ourselves to the triangle ratio efinition of the trig functions is that the possible angles are limite to the range θ [0, π ] raians or θ [0, 90] egrees. To remove this limitation, mathematicians extene the efinition of the trigonometric functions to a wier omain via the unit circle. θ Trigonometry Review - 4 How oes the circle efinition lea to the trigonometric ientity sin (θ) + cos (θ) =? Trigonometry Review - 5 Show how the circle an triangle efinitions efine the same values in the first quarant of the unit circle. Sine an Cosine as Oscillating Functions Sine an Cosine as Functions - Despite the geometric source of the trigonometric functions, they are use more commonly in biology an many other sciences as because their perioicity an oscillatory shapes. For many cyclic behaviours in nature, trigonometric functions are a natural first choice for moeling. It is useful to unerstan both efinitions of trig functions (circle an triangle) as sometimes one is more helpful than the other for a particular task.
Sine an Cosine as Functions - Sine an Cosine as Functions - 3 Question iagrams? 4 0 8 6 4 4 The graph of y = 0 + 4 cos(x) is shown in which of the following 4 0 8 6 4 Perio an Phase How can you fin the perio of the function cos(ax)? 6 A B 0 8 6 4 0 8 6 4 C D Show the amplitue an the average on the correct graph. Sine an Cosine as Functions - 4 Sine an Cosine as Functions - 5 How can you reliably etermine 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 calle the phase of the function.) Consier the graph of the function y = 5 + 8 cos(π(x )). following properties of the function: amplitue What are the perio average phase
Sine an Cosine as Functions - 6 Sketch the graph on the axes below. Inclue at least one full perio of the function. More complicate amplitues Non-Constant Amplitues - In the form y = A + B cos(cx + D), the B factor sets the amplitue. In many interesting cases, however, that amplitue nee not be constant. Sketch the graph of y = 5, an the graph of y = 5 cos(x) on the axes below. Non-Constant Amplitues - Sketch the graph of y = x, an the graph of y = x cos(πx) on the axes below. Use only x 0 Non-Constant Amplitues - 3 Use your intuition to sketch the graph of y = e x cos(πx) on the axes below.
Derivatives of Trigonometric Functions - Derivatives of Trigonometric Functions - Derivatives of Trigonometric Functions From Section 3.5 Having covere the graphs an properties of trigonometric functions, we can now review the erivative formulae for those same functions. The erivation of the formulas for the erivatives of sin an cos are an interesting stuy in both limits an trigonometric ientities. For those who are intereste, many such erivations can be foun on the web. However, it is in some ways more useful to erive the formula in a graphical manner. Below is a graph of sin(x). Use the graph to sketch the graph of its erivative. 3 π / π π/ 0 π/ π 3 π/ 3 π / π π/ 0 π/ π 3 π/ For example, http://www.math.com/tables/erivatives/more/trig.htm#sin From this sketch, we have evience (though not a proof) that Theorem x sin x = Derivatives of Trigonometric Functions - 3 Derivatives of Trigonometric Functions - 4 Most stuents will also be familiar with the other erivative rules for trig functions: cos(x) = sin(x) x x tan(x) = sec (x) sec(x) = sec(x) tan(x) x csc(x) = csc(x) cot(x) x x cot(x) = csc (x)
Derivatives of Trigonometric Functions - 5 Derivatives of Trigonometric Functions - 6 Prove the secant erivative rule, using the efinition sec(x) = cos(x) other erivative rules. an the Question: Fin the erivative of 4 + 6 cos(πx + ). 4 6 sin(πx + ) (πx). 6 cos(πx + ) (πx) 3. 6 sin(πx + ) (πx) 4. 6 sin(πx + ) (πx + ) 5. 6 sin(πx) Inverse Trig Functions - Inverse Trig Functions - Inverse Trig Functions From Section.5, 3.6 In aition to the 6 trig functions just seen, there are 6 inverse functions as well, though the inverses of sine, cosine, an tangent are the most commonly use. Sketch the graph of sin(x) on the axes below What is the omain of arcsin(x)? What is the range of arcsin(x)? On the same axes, sketch the graph of arcsin(x), or sin x, or the inverse of sin(x).
Derivative of arcsin - Derivative of arcsin - Derivative of arcsin Simplify sin(arcsin x) Solve for formula arcsin x, an simplify the resulting expression by means of the x which is vali if θ [ π, π ]. cos θ = sin θ, Differentiate both sies of this equation, using the chain rule on the left. You shoul en up with an equation involving arcsin x. x x arcsin x =