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# Criswell Calculus

### Definition of Derivative (Chapter 2)

Question | Answer |
---|---|

Definition: The Intermediate Value Theorem (IVT) | If "f" is a continuous function on the closed interval [a,b] and K falls between f(a) and f(b), then there must exist at least one c on the open interval (a,b) where f(c) =k |

Definition: Average Rate of Change | As a formula: {F(b) –F(a)} / (b –a), which would be the same as the definition of slope m = (y2 –y1) / (x2 – x1). |

Sum formula for sine | sin (x + h) = sin(x)cos(h) + sin(h)cos(x) |

Sum formula for cosine | cos (x + h) = cos(x)cos(h) – sin(x)sin(h) |

Definition: Derivative of a function | "Global Derivative" Lim h–>0 [ f (x+h) – f(x) ] / h |

What does it mean to take the derivative of a function? | Instantaneous Rate of Change. Slope of the tangent line for any point on the curve. |

Relationship of Derivatives on Projectiles | Displacement or Position Function: s(t), Velocity: s '(t) = v(t), Acceleration: s '' (t) = v ' (t) = a(t). |

Criteria for Standard Form Linear Equations | 1) Must be in "Ax + By = C" form 2) where A, B, & C are integers 3) and "A" must be a positive value m = –A / B or first number over the second and "change the sign". |

Linear Forms | Slope - y–intercept: y = mx + b or y = (∆y / ∆x) x + b. Standard: Ax + By = C Point–Slope: (y–k) = m (x–h) |

Derivative at a Point | "Disposable Derivative". Lim x–>c [ f (x) – f(c) ] / (x – c) |

Mean Value Theorem: Finding x -value where m-tangent = m-secant | States that if a function f is continuous on the closed interval [a,b], and differentiable on the open interval (a,b), then there exists at least point c in the interval (a,b) such that f'(c) is equal to the average rate of change over [a,b]. |

Average VS Instantaneous Rates of Change | An Average rate of change must be arrived at through MULTIPLE points (a secant) whereas an Instantaneous rate uses but one (a tangent) |

Estimating the Instantaneous Rate of Change | The best estimate from a table of values will come from using values that are a little under and a little over the desired target. Example f' (5) ≈ m sec =[f (5.5) – f (4.5) ] / (5.5 –4.5) even if f(5) is known it should NOT be used in the estimate! |

Understanding the terminology for "Instantaneous velocity" | When being asked about instantaneous velocity, this is a translation for instantaneous rate of change of displacement. S'(t) = v(t). This is a direct substitution into the velocity function. DO NOT MOVE a level! |

Understanding the terminology for "Instantaneous acceleration" | When being asked about instantaneous acceleration, this is a translation for instantaneous rate of change of velocity. S''(t) = v'(t) = a(t). This is a direct substitution into the acceleration function. |

Understanding the terminology for "Instantaneous rate of change" | When being asked about Instantaneous rate of change with regard to a function. One MUST MOVE DOWN a level! Ex1: Instantaneous rate of change of y (x) = y' (x) Ex2: Instantaneous rate of change of h '' (x) = h'''(x) |

Understanding the terminology for "Average velocity" | When being asked about average velocity. One MUST MOVE UP a level! average velocity of = [s (b) – s (a)] / ( b – a) Calculation is on S (t) from [ a ,b ] |

Understanding the terminology for "Average acceleration" | When being asked about average acceleration. One MUST MOVE UP a level! average acceleration of = [v (b) – v (a)] / ( b – a) Calculation is on v (t) from [ a ,b ] |

Understanding the terminology for "Average rate of change" | When being asked about average rate of change with regard to a function. DO NOT MOVE a level! Ex1: average rate of change of r ' (x) = [r ' (b) – r ' (a)] / ( b – a) Ex2: average rate of change of k '' (x) = [k '' (b) – K '' (a)] / ( b – a) |

Understanding the terminology for "Relative Max or Relative Min" | From the perspective of displacement S(t), max/min is found when S'(t) = v(t) = 0. One MUST MOVE DOWN a level and set equal to zero. Ex. Any function h''(x), One MUST MOVE DOWN a level and set equal to zero. So h'''(x)=0 |