Chapter 4 section 5

Excel Companion Chapter 4 Section 5
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Difference Quotient	*DiffQuot.XLS*

Open the file DiffQuot.xls

DiffQuot.XLS

The first derivative of y = f(x) is also known as the derived function or slope function and is denoted by y = f ¢ (x). A rough approximation to the graph of this derived function can be obtained from the values of the difference quotient when h is very small. The smaller the value of h the better the approximation to the actual derivative. Many graphing calculators use this approach to produce approximate first derivatives. The most common built-in algorithm for graphing calculators actually uses the symmetric difference with a default value of h = 0.001.

QUESTIONS:

1.	For each of the quadratic functions a) b) and c) draw a rough sketch of the function and its approximate derived function and record your observations.
	a) f(x) = x²
	b) f(x) = 5x² - 4x -3
	c) f(x) = -2x² + 6x + 3
	d) Make a conjecture about the derivative of any quadratic function
2.	For each of the cubic functions a) b) and c) draw a rough sketch of the function and its approximate derived function and record your observations.
	a) f(x) = x³
	b) f(x) = x³ + 4
	c) f(x) = - x³+ 6x² + 4
	d) Make a conjecture about the derivative of any cubic function
3.	a) The domain of the natural log function f(x) = LN(x) is all non-negative real numbers. Let xmin be 0.1 and sketch a rough graph of f(x) and its approximate derived function. Record your observations
	b) Which of the six functions in tangents.xls does the derived function resemble
4.	For each of the following exponential functions, examine the graph of y = f(x) and its approximate derived function and record your observations.
	a) f(x) = 2^x
	b) f(x) = 3^x
	c) f(x) = EXP(x)
	d) Make a conjecture about the derivative of EXP(x)

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