MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics §4.3 Exp & Log Derivatives

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §4.2 → Logarithmic Functions  Any QUESTIONS About HomeWork §4.2 → HW

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 3 Bruce Mayer, PE Chabot College Mathematics §4.3 Learning Goals  Differentiate exponential and logarithmic functions  Examine applications involving exponential and logarithmic derivatives  Employ logarithmic differentiation

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 4 Bruce Mayer, PE Chabot College Mathematics Derivative of ex  For any Real Number, x  Thus the e x fcn has the unusual property that the derivative of the fcn is the ORIGINAL fcn The proof of this is quite complicated. For our purposes we treat this as a formula –For a good proof (in Appendix) see:  D. F. Riddle, Calculus and Analytical Geometry, Belmont, CA, Wadsworth, 1974, pp

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 5 Bruce Mayer, PE Chabot College Mathematics Derivative of ex  Using the “repeating” nature of d(e x )/dx  Meaning of Above: for any x-value, say x = 1.9, All of these y-related quantities are equal at e 1.9 = The y CoOrd: The Slope: The ConCavity:

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 6 Bruce Mayer, PE Chabot College Mathematics Example  ex Derivative  Differentiate:  Using Rules Product Power e x

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 7 Bruce Mayer, PE Chabot College Mathematics Chain Rule for eu(x)  If u(x) is a differentiable function of x then  Using the e x derivative property

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 8 Bruce Mayer, PE Chabot College Mathematics Example  Tangent Line  Find the equation of the tangent line at x = 0 for the function:  SOLUTION:  Use the Point-Slope Line Eqn, y-y AP = m(x-x AP ), with Anchor Point, (x AP,y AP ): Slope at the Anchor Point:

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 9 Bruce Mayer, PE Chabot College Mathematics Example  Tangent Line  Find Slope at x = 0  Let:  Then:  Thus:  And by Chain Rule

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 10 Bruce Mayer, PE Chabot College Mathematics Example  Tangent Line  Then m at x = 0  Using m and the Anchor-Point in the Pt-Slope Eqn  Convert Line-Eqn to Slope-Intercept form

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 11 Bruce Mayer, PE Chabot College Mathematics Example  Tangent Line  Tangent Line at (0,1) Graphically

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 12 Bruce Mayer, PE Chabot College Mathematics Derivative of ln(x) = loge(x)  For any POSITIVE Real Number, x  Thus the ln(x) fcn has the unusual property that derivative Does NOT produce another Log The proof of this is quite complicated. For our purposes we treat this as a formula –For a good proof (in Appendix) see:  D. F. Riddle, Calculus and Analytical Geometry, Belmont, CA, Wadsworth, 1974, pp

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 13 Bruce Mayer, PE Chabot College Mathematics Example  ln Derivative  Find the Derivative of:  Using Rules Quotient Power ln(x)

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 14 Bruce Mayer, PE Chabot College Mathematics Chain Rule for ln(u(x))  If u(x)> 0 is a differentiable function of x then  Using the ln(x) derivative property

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 15 Bruce Mayer, PE Chabot College Mathematics Derivative of ax & loga(x)  For Base a with a>0 and a≠1, then for ALL x:  For Base a with a>0 and a≠1, then for ALL x>0:  Prove Both on White/Black Board

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 16 Bruce Mayer, PE Chabot College Mathematics Example  Revenue RoC  The total number of hits (in thousands) to a website t months after the beginning of 1996 is modeled by  The Model for the weekly advertising revenue in ¢ per hit:  Use the Math Models to determine the daily revenue change at the beginning of the year 2005

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 17 Bruce Mayer, PE Chabot College Mathematics Example  Revenue RoC  SOLUTION:  The rate of change in Total Revenue, R(t), is the Derivative of the Product of revenue per hit and total hits:

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 18 Bruce Mayer, PE Chabot College Mathematics Example  Revenue RoC  Thus  Next find t in months for 1996→2005  Then the rate derivative at t = 108 mon  A units analysis

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 19 Bruce Mayer, PE Chabot College Mathematics Example  Revenue RoC  The units on H are kHits, and units on r are ¢/Hit. The units on time were months so the derivative has units k¢/mon. Convert to $/mon:  STATE: at the beginning of 2005 the website was making about $ LESS each month that passed.

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 20 Bruce Mayer, PE Chabot College Mathematics Helpful Hint  Log Diff  Logarithmic Differentiation  Some derivatives are easier to calculate by first take the natural logarithm of the expression Next judiciously use the log rules then take the derivative of both sides of the equation finally solve for the derivative term

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 21 Bruce Mayer, PE Chabot College Mathematics Example  Using Log Diff  Using logarithmic differentiation to find the df/dx for:  SOLUTION:  Computing the derivative directly would involve the repeated use of the product rule (not impossible, but very tedious)  Instead, use properties of logarithms to first expand the expression

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 22 Bruce Mayer, PE Chabot College Mathematics Example  Using Log Diff  Let y = f(x) →  Then take the natural logarithm of both sides:  Use the Power & Log Rules  Now Take the Derivative of Both Sides

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 23 Bruce Mayer, PE Chabot College Mathematics Example  Using Log Diff  By the Chain Rule  Then  Or This is a form of Implicit Differentiation; Need to algebraically Isolate dy/dx

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 24 Bruce Mayer, PE Chabot College Mathematics Example  Using Log Diff  Solving for dy/dx  Recall  Thus This result would have much more difficult to obtain without the use of the Log transform and implicit differentiation

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 25 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §4.3 P76 → Per Capita Growth P90 → Newtons Law of (convective) Cooling – Requires a Biot Number* of Less than 0.1 Bi → INternal Thermal Resistance EXternal Thermal Resistance *B. V. Karlekar, R. M. Desmond, Engineering Heat Transfer, St. Paul, MN, West Publishing Co., 1977, pp

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 26 Bruce Mayer, PE Chabot College Mathematics All Done for Today For PHYS4A Students From RigidBody Motion-Mechanics

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 27 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics Appendix –

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 28 Bruce Mayer, PE Chabot College Mathematics ConCavity Sign Chart abc −−−−−−++++++−−−−−− x ConCavity Form d 2 f/dx 2 Sign Critical (Break) Points InflectionNO Inflection Inflection

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 29 Bruce Mayer, PE Chabot College Mathematics Summary of Log Rules  For any positive numbers M, N, and a with a ≠ 1, and whole number p Product Rule Power Rule Quotient Rule Base-to-Power Rule

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 30 Bruce Mayer, PE Chabot College Mathematics Change of Base Rule  Let a, b, and c be positive real numbers with a ≠ 1 and b ≠ 1. Then log b x can be converted to a different base as follows:

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 31 Bruce Mayer, PE Chabot College Mathematics Derive Change of Base Rule  Any number >1 can be used for b, but since most calculators have ln and log functions we usually change between base-e and base-10

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 32 Bruce Mayer, PE Chabot College Mathematics Prove d(e x )/dx =e x –D. F. Riddle, Calculus and Analytical Geometry, Belmont, CA, Wadsworth, 1974, pp

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 33 Bruce Mayer, PE Chabot College Mathematics Prove d(e x )/dx =e x

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 34 Bruce Mayer, PE Chabot College Mathematics  D. F. Riddle, Calculus and Analytical Geometry, Belmont, CA, Wadsworth, 1974, pp

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 35 Bruce Mayer, PE Chabot College Mathematics Prove:

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 36 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 37 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 38 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 39 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 40 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 41 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 42 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 43 Bruce Mayer, PE Chabot College Mathematics