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Applied Calculus (MAT 121) Dr. Day Thursday, March 22, 2012

Applied Calculus (MAT 121) Dr. Day Thursday, March 22, 2012. Derivatives of Log Functions (5.5) Exponential Models (5.6) Remember Your Derivative Rules! Derivative Rule for Exponential Functions Derivative Rule for Logarithmic Functions Using and Applying these Derivatives.

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Applied Calculus (MAT 121) Dr. Day Thursday, March 22, 2012

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  1. Applied Calculus (MAT 121)Dr. Day Thursday, March 22, 2012 Derivatives of Log Functions (5.5) Exponential Models (5.6) • Remember Your Derivative Rules! • Derivative Rule for Exponential Functions • Derivative Rule for Logarithmic Functions • Using and Applying these Derivatives MAT 121

  2. Exponential Functions Big Ideas • What are Exponential Growth and Exponential Decay? • What sort of applications use exponential functions as their mathematical models? • What algebra do we need to remember to work with exponential functions? • How do we undo exponentiation? We need the inverse function, called the logarithm function. • Where does calculus fit in to all of this? MAT 121

  3. Compound Interest Periodic Compounding • A: amount in account • P: amount of original deposit (principal) • i: annual interest rate, expresses as a decimal value • n: number of compounding periods per year • t: number of years the principal remains in account Continuous Compounding • A: amount in account • P: amount of original deposit (principal) • r: annual interest rate, expresses as a decimal value • t: number of years the principal remains in account MAT 121

  4. Review: Derivative Rules • Power Rule • Product Rule • Quotient Rule • Chain Rule MAT 121

  5. Derivatives of Exponential Functions • For y = ex: • For y= kex: • For y= eu(x): MAT 121

  6. Derivatives of Exponential Functions y= ex y= 3ex y= e-2x y= 5x3 + e5x y= (1+e2x)/(e-3x-x) MAT 121

  7. Using These Derivatives • Determine all intervals where the function f(x) = x2e-x is increasing and where it is decreasing. • Find the absolute extrema of the function. h(x) = ex2 - 4 on [-2,2]. • The percentage of alcohol in a person's bloodstream thrs after drinking 8 fluid oz of whiskey is modeled by the equation. A(t) = 0.29te-0.3t, 0 ≤ t ≤ 12. How fast is the percentage of alcohol in a person's bloodstream changing after 1/2 hr? After 5 hr? Round your answers to four decimal places. MAT 121

  8. Derivatives of Logarithmic Functions • For y = ln(x): • For y= kln(x): • For y= ln(u(x)): MAT 121

  9. Derivatives of Logarithmic Functions y= ln(x) y= ln(3x) y= x2ln(3x-1) y= e5xln(5x) y= (ln(3x))/(3ln(x)) MAT 121

  10. Exponential Growth & Decay Suppose that a quantity Q(t) is modeled by the exponential growth function Q(t) = 390e0.07t, where t is measured in minutes. Answer the following questions. • What is the growth constant? • What is the initial quantity (t = 0)? • Calculate Q(20). • How fast is the quantity Q changing at time t = 10 minutes? MAT 121

  11. Exponential Growth & Decay A road grader was purchased 6 years ago for $625,000. Today its resale value is $400,000. Suppose the value of the road grader decreases exponentially: • Create an exponential function V(t) that models the resale value of the grader, where t is measured in years and t = 0 represents the time the equipment was purchased. • Use your model to determine the resale value of the road grader 4 years form now. • What was the rate of change of the resale value of the grader exactly 5 yearsafter it was purchased? • How long, from the date of purchase, will it be before the road grader first has a value less that $120,000? MAT 121

  12. Exponential Growth & Decay The growth rate of Escherichia coli, a common bacterium found in the human intestine, is proportional to its size. Under ideal laboratory conditions, when this bacterium is grown in a nutrient broth medium, the number of cells in a culture doubles approximately every 30 min. • If the initial population is 500 cells, determine the function Q(t) that expresses the growth of the number of cells of this bacterium as a function of time t (in minutes). • How long will it take for the initial population of 500 cells to grow to 200,000 cells? MAT 121

  13. Exponential Growth & Decay Phosphorus-32 (P-32) has a half-life of 14.2 days. If 350 g of this substance are present initially, find the amount Q(t) present after t days. (Round your growth constant to four decimal places.) MAT 121

  14. Assignments WebAssign 5.6 due tonight! Test #4: Friday, March 23, 2012: • 14 multiple-choice questions (6 pts each  84 pts total) • 2 open-response questions (3 parts each, 4 pts each part  24 pts total) MAT 121

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