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Applied Calculus (MAT 121) Dr. Day Monday, March 19 , 2012

Applied Calculus (MAT 121) Dr. Day Monday, March 19 , 2012

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Applied Calculus (MAT 121) Dr. Day Monday, March 19 , 2012

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  1. Applied Calculus (MAT 121)Dr. Day Monday, March 19, 2012 Derivatives of Exponential Functions(5.4) and Log Functions (5.5) • 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. Assignments WebAssign 5.4 due tonight! 5.5 due Wednesday night 5.6 due Thursday night Test #4: Friday, March 23, 2012. MAT 121