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DERIVATIVES

How To Solve Them The Right Way. DERIVATIVES. The Original Equation. Endpoints [-10, 10]. Step 1: Take the 1 st Derivative. Endpoints [-10, 10]. Take the 1 st derivative. Step 2: Set f’(x)= 0. Endpoints [-10, 10]. Solve It. These are the critical points!. ( 3x-16 )( x-8 )

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DERIVATIVES

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  1. How To Solve Them The Right Way DERIVATIVES

  2. The Original Equation Endpoints [-10, 10]

  3. Step 1: Take the 1st Derivative Endpoints [-10, 10] Take the 1st derivative

  4. Step 2: Set f’(x)= 0 Endpoints [-10, 10] Solve It These are the critical points! (3x-16)(x-8) x= 16/3 x= 8

  5. Step 3: Make 1st Derivative Table Endpoints [-10, 10] -10 16/3 6 8 10 + 0 - 0 + f ’(x) f (x) -10 16/3 20/3 8 10 (-10)^3 – 20(-10)^2 +128(-10) – 280 = -560 (16/3)^3 – 20(16/3)^2 +128(16/3) – 280 = -14.518 Point of Inflection (20/3)^3 – 20(20/3)^2 +128(20/3) – 280 = -19.258 See Slide 9 for further details (8)^3 – 20(8)^2 +128(8) – 280 = -24 (10)^3 – 20(10)^2 +128(10) – 280 = 0

  6. Step 4: Determine Maxs & Mins Endpoints [-10, 10] -10 16/3 6 8 10 + 0 - 0 + f ’(x) max min There is a max at x= 16/3 because f ‘ (x) changes from positive 0 negative There is a min at x= 8 because f ‘ (x) changes from negative 0 positive

  7. Step 4: Determine Maxs & Mins Endpoints [-10, 10] f (x) -10 16/3 20/3 8 10 (-10)^3 – 20(-10)^2 +128(-10) – 280 = -560 GLOBAL MIN (16/3)^3 – 20(16/3)^2 +128(16/3) – 280 = -14.518 MAX (20/3)^3 – 20(20/3)^2 +128(20/3) – 280 = -19.258 POINT OF INFLECTION (8)^3 – 20(8)^2 +128(8) – 280 = -24 MIN (10)^3 – 20(10)^2 +128(10) – 280 = 0 GLOBAL MAX x= -10 is an endpoint on the left and f ‘(x) to its right is positive :. x= -10 is a min x= 10 is an endpoint on the right and f ‘(x) to its left is positive :. x= 10 is a max

  8. The 1st Derivative f ‘(x) f(x)

  9. Step 5: 2nd Derivative Test Endpoints [-10, 10] To Find the Point(s) of Inflection 6x – 40 = 0 x= 20/3 f “(x) = 6x – 40 f “(x) 16/3 8 6(16/3) – 40 = -8 < 0 Max 6(8) – 40 = 8 > 0 MIN -10 20/3 10 - 0 + f “(x) Concave down

  10. The 2nd Derivative Endpoints [-10, 10]

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