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Problem of the Day

Problem of the Day. Let f be a differentiable function such that f(3) = 2 and f '(3) = 5. If the tangent line to the graph of f at x = 3 is used to find an approximation to a zero of f , that approximation is. A) 0.4 B) 0.5 C) 2.6 D) 3.4 E) 5.5. Problem of the Day.

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Problem of the Day

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  1. Problem of the Day Let f be a differentiable function such that f(3) = 2 and f '(3) = 5. If the tangent line to the graph of f at x = 3 is used to find an approximation to a zero of f, that approximation is A) 0.4 B) 0.5 C) 2.6 D) 3.4 E) 5.5

  2. Problem of the Day Let f be a differentiable function such that f(3) = 2 and f '(3) = 5. If the tangent line to the graph of f at x = 3 is used to find an approximation to a zero of f, that approximation is A) 0.4 B) 0.5 C) 2.6 D) 3.4 E) 5.5 Point (3, 2) Slope = 5 Tangent y - 2 = 5(x - 3) Thus y = 5x - 13 To find zero 0 = 5x - 13 x = 2.6

  3. If h(x) = f(x)g(x) what is the derivative?

  4. If h(x) = f(x)g(x) what is the derivative? lim Δx 0 f(x + Δx)g(x + Δx) - f(x)g(x) Δx add a well chosen zero f(x+Δx)g(x+Δx) + f(x+Δx)g(x) - f(x+Δx)g(x) - f(x)g(x) Δx

  5. If h(x) = f(x)g(x) what is the derivative? lim Δx 0 f(x + Δx)g(x + Δx) - f(x)g(x) Δx add a well chosen zero f(x+Δx)g(x+Δx) - f(x+Δx)g(x) + f(x+Δx)g(x) - f(x)g(x) Δx f(x+Δx)(g(x+Δx) - g(x)) + g(x)(f(x+Δx) - f(x)) lim Δx 0 Δx

  6. g(x)(f(x+Δx) - f(x)) lim Δx 0 Δx If h(x) = f(x)g(x) what is the derivative? lim Δx 0 f(x + Δx)g(x + Δx) - f(x)g(x) Δx add a well chosen zero f(x+Δx)g(x+Δx) - f(x+Δx)g(x) + f(x+Δx)g(x) - f(x)g(x) Δx f(x+Δx)(g(x+Δx) - g(x)) + g(x)(f(x+Δx) - f(x)) lim Δx 0 Δx lim Δx 0 + f(x+Δx)(g(x+Δx) - g(x)) Δx

  7. If h(x) = f(x)g(x) what is the derivative? g(x) (f(x+Δx) - f(x)) lim Δx 0 lim Δx 0 + f(x+Δx) (g(x+Δx) - g(x)) Δx Δx Evaluate limits f(x) g'(x) + g(x) f '(x)

  8. If h(x) = f(x)g(x) what is the derivative? Product Rule f(x) g'(x) + g(x) f '(x) 1st times derivative of second + 2nd times derivative of 1st (Rule extends to cover more than 2 factors) if j(x) = f(x)g(x)h(x) then j'(x) = f '(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x)

  9. Find the derivative of y = 2xcos x - 2sin x

  10. Find the derivative of y = 2xcos x - 2sin x y' = 2x(-sinx) + cos x(2) - 2cos x y' = -2xsin x

  11. Quotient Rule (Proof is in textbook) if h(x) = f(x) g(x) then h'(x) = g(x)f '(x) - f(x)g'(x) (g(x))2

  12. Quotient Rule (Proof is in textbook) Hi Lo if h(x) = f(x) g(x) then h'(x) = g(x)f '(x) - f(x)g'(x) (g(x))2 Lo d Hi - Hi d Lo Lo Lo

  13. Quotient Rule Find the derivative of 7x2 - 4 3x4 - 2x

  14. Quotient Rule Find the derivative of 7x2 - 4 3x4 - 2x (3x4 - 2x)(14x) - (7x2 - 4)(12x3 - 2) (3x4 - 2x)2

  15. Find the derivative of x2 + 3x 6 Find the derivative of 9 5x2 Find the derivative of -3(3x - 2x2) 7x

  16. Caution! Not every quotient needs the quotient rule. Find the derivative of x2 + 3x 6 y = 1 (x2 + 3x) 6 y' = 1 (2x + 3) 6

  17. y = 9 x-2 5 Find the derivative of 9 5x2 y' = 9 (-2x-3) 5 y' = -18 5x3

  18. Find the derivative of -3(3x - 2x2) 7x y = -3(3 - 2x) 7 y' = -3(-2) 7 y' = -6 7

  19. Trig Derivatives d tan x = sec2x dx d sec x = secx tanx dx d cot x = - csc2x dx d csc x = - cscx cotx dx

  20. Find the derivative of y = x2csc x

  21. Find the derivative of y = x2csc x y' = x2(-cscx cotx) + 2x(csc x) = (x cscx)(-x cotx + 2) How do you know when you have finished? Combine like terms and remove negative exponents.

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