Business Calculus Exponentials and Logarithms

 3.1 The Exponential Function Know your facts for 1.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. Evaluate exponential functions by calculator. 4.Solve exponential functions using the logarithm. 5. Differentiate : 6. Differentiate exponential functions using the sum/difference, coefficient, product, quotient, or chain rule. 7.Find relative extrema, absolute extrema. 8.Use in marginal analysis or related rates, and interpret.

 3.2 Logarithmic Function Know your facts for 1.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:

3. Evaluate logarithmic functions by calculator. 4.Solve logarithmic functions using the exponential. 5.Properties of logarithms:

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.

 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.

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.  3.3 & 3.4 Growth and Decay Models  Uninhibited Growth

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  Uninhibited 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  Limited Growth/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 = ce kx. Instead, it uses y = ab x.  Modeling Growth and Decay