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## Applications of Derivatives

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**Applications of Derivatives**Section 4.1 Section 5.2**Applications of Derivatives**• Derivatives allow you to sketch the shape of functions**Applications of Derivatives**• Ex: Amount of cargo unloaded at a port related to the number of trucks**Applications of Derivatives**• Sketch the function c(w) based on the following: c(0) = 200 c(5) = 176 c(20) = 121 c’(0) = -50 c’(5) = -44 c’(20) = -30**Applications of Derivatives**• Derivatives allow you to approximate functions**Applications of Derivatives**• Suppose that for the function c(w), c(10) = 155 and c’(10) = -39. What is the approximate value of c(20)?**Extreme Points**0 + slope - slope**Extreme Points**Population of Cleveland**Extreme Points**• Conclusions • At the minimum/maximum values of a function, the value of the derivative is 0. • At the inflection points of a function, the value of the derivative reaches a minimum/maximum.**Extreme Points**• Finding roots • Easy for linear, quadratic • Hard for higher order polynomials, other function Y= GRAPH CALC 2: zero**Extreme Points**• In-Class • Find the maxima and minima for the following functions • 0.04x3 - 0.88x2 + 4.81x +12.11 • 0.0004x4 – 0.007x3 + 0.03x2 – 0.035x + 10**Extreme Points**• Cost of production • How many machines are needed to minimize the cost per unit?**Extreme Points**• Fit a quadratic model to the data**Extreme Points**• How many machines are needed to minimize the cost per unit?**Extreme Points**• How many machines are needed to minimize the cost per unit? • The number that sets c’(m) = 0 (root)**Extreme Points**• Revenue over time • In what month was revenue maximized?**Extreme Points**• Fit a quartic model to the data**Extreme Points**• In what month was revenue maximized?**Extreme Points**• In what month was revenue maximized? • Find the 3 numbers that set r’(t) = 0 Y= GRAPH CALC 2: zero**Extreme Points**• In what month was revenue maximized? • Find the 3 numbers that set r’(t) = 0**Extreme Points**• In-Class