1. Explain in detail where the formula for the difference quotient comes from now that you know. Include all appropriate terminology (secant line, tangent line, h/delta x, etc.).
The difference quotient is:
In order to derive the difference quotient, we must first keep in mind a couple things. A tangent line touches the graph once while a secant graph touches the graph at two points. The first point is (x, f(x)) and the second point is (x+h, f(x+h)). The second point is farther than x and also, it is h units away from x (keep in mind that h is also refered to as delta x). To find where the difference quotient comes from, we would then plug these points into the slope formula, which is (y2-y1/x2-x1). After plugging these points into the equation, you will notice that some stuff cancels; this then leaves the denominator as just being h. When the problem completely simplified, you discover that you get the difference quotient. The difference quotient helps us find the derivative, or the slope of the tangent line.
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