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Chapter 5-Derivatives of Exponential and Trigonometric Functions

Chapter 5-Derivatives of Exponential and Trigonometric Functions. By Jeffrey Kim, Chris Bayley, Jacqueline Tennant, Fredy Valderrama. Agenda. Review of Pre-requisite Skills Derivative of General Exponential Functions Derivative of the Exponential Function

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Chapter 5-Derivatives of Exponential and Trigonometric Functions

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  1. Chapter 5-Derivatives of Exponential and Trigonometric Functions By Jeffrey Kim, Chris Bayley, Jacqueline Tennant, FredyValderrama

  2. Agenda • Review of Pre-requisite Skills • Derivative of General Exponential Functions • Derivative of the Exponential Function • Optimization Problems with Exponential Functions • Derivatives of Sinusoidal Functions

  3. Things you should know: Exponent Laws

  4. Things you should know: LogarithmLaws

  5. More Things You Should Know ()= ()=f()() ’()=f’()()+f()’() 1. Product rule ’()=( ’()=)()+ ’()= 2. Quotient rule ()= ’()= f()= f’()= f’()= 3. Chain rule ()= f(()) ’()=f’(()) ’() f()=6 f’()=(36-30) f’()=6(6-5)

  6. f(x) = ex f’(x) = ex g(x) = eh(x) g’(x) = eh(x) * h’(x)

  7. K/U question Differentiate the following function:

  8. Consider the function f(x) = 3x. 5.2: Derivative of the General Exponential Function, y = bx Key Points: • f’(x) is a vertical stretch or compression of f(x), dependent on the value of b • the ratio f’(x)/f(x) is a constant and is equal to the stretch/compression factor Derivative of f(x) = bx: f’(x) = (ln b) * bx Derivative of f(x) = bg(x): f’(x) = bg(x) * (ln b) * g’(x)

  9. Graphing f(x) and f’(x): f(x) = 3x f’(x) = 1.10 • 3x

  10. Examples Ex 1 find the derivative of f(x) a) f(x) = 8x b) f(x) = 34+2

  11. QuestionforTest Knowledge • Find the derivative of f(x) a) f(x) = 8x b) f(x) = 5 2x^2 – 3x + 10

  12. QuestionforTest Application You purchased a new car for $16,000. the value of the car after t years is given by the function, V(t) where t is the number of years after the purchased and v(t) is the value of the car in dollars V(t)=16000(0.78)t • Determine the value of the car after the first year. • Find the rate of change when t=1 • Interpret the results.

  13. 5.3Optimization Problems with Exponential Functions

  14. Algorithm for Solving Optimization Problems: • Understand the problem, and identify quantities that can vary. Determine a function in one variable that represents the quantity to be optimized. • Whenever possible, draw a diagram, labelling the given and required quantities. • Determine the domain of the function to be optimized, using the information given in the problem. • Use the algorithm for extreme values to find the absolute maximum or minimum value in the domain. • Use your result for step 4 to answer the original problem.

  15. ApplicationQuestion Jack, given $24 from his parents, wants to bake cookies for profit. The cost of baking 1 batch of 24 cookies is $12. If his revenues from the sale of these cookies are modeled by f(x) = e3x – 150x, and Jack must sell all his cookies after baking before baking more cookies, find the number of cookies Jack must sell after his initial round of baking that maximizes his profits.

  16. Derivatives of Sinusoidal Functions

  17. Key Concepts • F(x)=sinx , then = cosx • F(x)=cosx, then =-sinx Composite sinusoidal functions • If y=sinf(x), then y’ = cosf(x)× f’(x) • If y=cosf(x) , then = -sinf(x) × f’(x)

  18. Knowledge Question Find the derivative of y = tan(x)

  19. Homework to review

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