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## Limits and Derivatives

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**Limits and Derivatives**2.8 The Derivative as a Function 2.8 The Derivative as a Function**Derivative**Last class, we considered the derivative of a function f at a fixed number a, 2.8 The Derivative as a Function**Derivative**Today, we want to let avary in order to create a function. To do so we replace ain our previous equation with x and get the derivative of f defined by 2.8 The Derivative as a Function**Interpretation of the Derivative**• Remember, the value of f ′(x) can be interpreted geometrically as the slope of the tangent line to the graph of the original function f at the point (x, f (x)). • We can use the graph of the function to sketch the derivative. 2.8 The Derivative as a Function**Example 1**• The graph of a function f is given below. Use it to sketch the graph of the derivative f '. 2.8 The Derivative as a Function**Example 1**• Solution 2.8 The Derivative as a Function**Try these on your own…**• Sketch the Derivative 2.8 The Derivative as a Function**Example 2**• Find f ′ and state the domain of f ′ if 2.8 The Derivative as a Function**Other Notations**All of the following mean the derivative of the function with respect to x. Newton’s Dot Notation: Leibniz: Lagrange: Euler: Prime Notation: 2.8 The Derivative as a Function**Other Notations**• If we want to indicate the value of a derivative dy/dx in Leibniz notation at a specific number a, we use the notation which is a synonym for f ′(a). 2.8 The Derivative as a Function**Other Derivatives**• Function – Position x • First derivative – Velocity • Second derivative – Acceleration • Third derivative - Jerk 2.8 The Derivative as a Function**Finding the Second Derivative**2.8 The Derivative as a Function**Example 3**• Find the first and second derivatives of the following function: 2.8 The Derivative as a Function**Book Example – Page 162 #1**• Use the graph to estimate the value of each derivative. Sketch the graph of f’. 2.8 The Derivative as a Function**Book Example – Page 162 #2**• Use the graph to estimate the value of each derivative. Sketch the graph of f’. 2.8 The Derivative as a Function**Book Example – Page 162 # 6, 8, & 10**• Trace or copy the graph of the given function f. Assuming that the axis have equal scales, sketch the graph of f’ below it. 2.8 The Derivative as a Function**Example**• Find the derivative of the function using the definition of its derivative. State the domain of the function and the domain of its derivative. 2.8 The Derivative as a Function**Book Example – Page 163 #33**• If , find f(x). • Check to see that your answer in (a) is reasonable by comparing the graphs of f and f. 2.8 The Derivative as a Function**Book Example – Page 164 #35**• The unemployment rate U(t) varies with time. The table from the Bureau of Statistics gives the percentage of unemployed in the US labor force from 1999 to 2008. • What is the meaning of U(t)? What are its units? • Construct a table of values for U(t). 2.8 The Derivative as a Function**Book Example – Page 164 #45**• The figure shows the graphs of three functions. One is the position function of a car, one is the velocity function of a car, and one is acceleration of the car. Identify each curve and explain your choices. 2.8 The Derivative as a Function**What’s next?**• So far we have: • Interpreted derivatives as slopes and rates of changes • Estimated derivatives of functions given by tables of values • Graphed derivatives of functions that are defined graphically • Used the definition of derivative to calculate the derivatives of functions defined by formulas. • Next time we will develop rules for finding the derivatives without having to use the definition directly. 2.8 The Derivative as a Function