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# Criswell Calculus

### Definition of Derivative (Chapter 2)

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
Created by: Troy.Criswell
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