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Business Calculus

Business Calculus. Exponentials and Logarithms. 3.1 The Exponential Function. Know your facts for Know the graph: A horizontal asymptote on the left at y = 0. Through the point (0,1) Domain: (-∞, ∞) Range: (0, ∞) Increasing on the interval (-∞, ∞) .

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Business Calculus

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  1. Business Calculus Exponentials and Logarithms

  2. 3.1 The Exponential Function Know your facts for Know the graph: A horizontal asymptote on the left at y = 0. Through the point (0,1) Domain: (-∞, ∞) Range: (0, ∞) Increasing on the interval (-∞, ∞) . 2. Use the graph to find limits:

  3. 3. Evaluate exponential functions by calculator. Solve exponential functions using the logarithm. 5. Differentiate : 6. Differentiate exponential functions using the sum/difference, coefficient, product, quotient, or chain rule. Find relative extrema, absolute extrema. Use in marginal analysis or related rates, and interpret.

  4. 3.2 Logarithmic Function Know your facts for Know the graph: A vertical asymptote below the x axis at x = 0. Through the point (1,0). Domain: (0, ∞) Range: (-∞, ∞) Increasing on the interval (0, ∞) . 2. Use the graph to find limits:

  5. 3. Evaluate logarithmic functions by calculator. Solve logarithmic functions using the exponential. Properties of logarithms:

  6. 6. Change of Base formula: 7. Differentiate : 8. Differentiate logarithmic functions using the sum/difference, coefficient, product, quotient, or chain rule. 9. Find relative extrema, absolute extrema. 10. Use in marginal analysis or related rates, and interpret.

  7. Logarithmic Differentiation A new way to differentiate functions that are products and quotients involves the properties of logarithms. If y = f (x) is a function which uses the product, quotient, or chain rules in combination, we can consider a new problem: Take the natural log of both sides ln(y) = ln(f (x)) Rewrite ln(f (x)) using properties of logs Differentiate both sides with respect to x Solve for dy/dx. Note: when we take the natural log of both sides, the derivative becomes implicit.

  8. 3.3 & 3.4 Growth and Decay Models • Uninhibited Growth Uninhibited growth is a function that grows so that the rate of change of output with respect to input is proportional to the amount of output. The formula for this is (for y output and x input). This can only be true if the function is , k > 0. In this exponential function, k represents the growth rate of y, and c represents the amount of y when x = 0.

  9. Uninhibited Decay Uninhibited decay is a function that declines so that the rate of change of output with respect to input is proportional to the amount of output. The formula for this is (for y output and x input). This is true if the function is , k < 0. In this exponential function, k represents the decay rate of y, and c represents the amount of y when x = 0. exponential exponential growth k > 0 decay k < 0

  10. Limited Growth/Decay Logistic growth is an example of a limited growth model. This function is a growth function if k > 0, and it is a decay function if k < 0. k > 0 k < 0

  11. Modeling Growth and Decay When analyzing information, we may be given data points instead of a function. We will make use of the regression capability of our calculator to find a function that approximates a set of data. Print the Regression Equation handout on blackboard to find a list of steps to create this function. Important Note: The exponential function used by the calculator is not y = cekx . Instead, it uses y = abx.

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