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Euler, Exponential, and Logistic Functions
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Classwork & Homework Only if you stay on task!!! FRQ 11 & 13
Rest of the FRQs All Multiple choices Only if you stay on task!!!
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Card 49: Steps for Euler’s Method
1.Create Chart in terms of “x”, “y”, and “dy/dx” 2.Use your initial x-value and your step size to fill out all of the x-values in the chart 3.Using the initial x-value and initial y-value, solve for dy/dx 4.Use equation ynew= yold + (Δx)dy/dx
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Chart x y dy/dx 1 2 2(2)+(1)=5 1.5 2+(0.5)5=4.5 ynew= yold+(Δx)dy/dx
dy/dx=2y+x F(1)=2 Step-size (Δx) =0.5 Find f(1.5) x y dy/dx 1 2 2(2)+(1)=5 1.5 2+(0.5)5=4.5
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Direct Variation Formula: y = k*x
Card 50: Relationship Between Direct Variation and Exponential Growth/Decay Direct Variation Formula: y = k*x “y” varies directly with “x”; as “x” increases, so does “y” Inverse Variation Formula: k = x*y “y” varies inversely with “x”; as “x” increases, “y” decreases and the opposite applies
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Card 50: Relationship Between Direct Variation and Exponential Growth/Decay
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Exponential Growth and Decay Formula: dy/dt = k*y or y = C*ekt
Card 50: Relationship Between Direct Variation and Exponential Growth/Decay Exponential Growth and Decay Formula: dy/dt = k*y or y = C*ekt When asked to find the “general solution” use the equation to the right C is initial value (constant) k is proportionality constant k>0 : exponential growth; k<0 : exponential decay
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Card 50: Relationship Between Direct Variation and Exponential Growth/Decay
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Card 51: Logistic Growth “M” represents the carrying capacity
A logistic growth model is a more realistic model which accounts for limiting factors and where growth rate is proportional to the amount present (P)
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Card 51: Logistic Growth
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Card 51: Logistic Growth
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The “S” curve or “S” function’s proper name is the Sigmoid Function
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