AP Calculus

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Reciprocals of Zero and Infinity

If f(x) equals 1 over g(x) and limit g(x) as x approaches c equals 0, then limit f(x) as x approaches c is infinite.
If f(x) equals 1 over g(x) and limit g(x) as x approaches c is infinite, then limit f(x) as x approaches c equals 0.
The same properties apply if [...]

Removable Discontinuity: Must be able to find the limit. The value of f(c) can be defined or redefined to make f continuous there. has a discontinuity at x = 3 because the denominator is zero there. It seems reasonable to say that the function is “continuous” everywhere else because the graph seems to [...]

The first lab explored the properties of derivatives and their relation to functions using tangent lines. For example, if the derivative at x=c is positive, then the slope is positive, meaning the function is increasing at x=c. If the derivative is negative, then the slope is negative, meaning the function is decreasing at x=c. If [...]

Today in class we did a gsp on tangent lines and derivatives.

For a increasing function à derivative is greater than 0.
For a decreasing functionà derivative is less than 0.
At high/low pts on the graph à derivative=0.

Derivative can tell you where the maximum and minimums are as well as where the function is increasing or decreasing.

If [...]

What I did learn from the exercises
•    I have learnt what the average rate of change between two points on a function looks like. And when one point approaches the other, average rate approaches instantaneous rate. Even more I have learnt what the instantaneous rate look like.
•    A tangent line is a line that interact [...]

The Getting Off on a Tangent lab showed a number of things. It first pointed out that the average rate of change between two points on a function is the slope of the secant line between the two points. Then as one point approaches the other, the average rate approaches the instantaneous rate. [...]

These labs further differentiated between instantaneous rate of change (derivative) and average rate of change. The gsp lab illustrated using a tangent line how the slope of the derivative increases as the function increases and decreases as the function decreases. This is especially helpful in cases where we are not able to graph the function [...]

Well ella summed the labs up very well. The first lab helped show me where we will be going, in describing the derivative in a graph and how the derivative is related to the the functution. its poositive when the function increasing, negative when the funtion is decresing, and zero at the appex of curve. [...]

From these labs I have learned about the differences between instantaneous rates and average rates. The second lab shows us about what can be inaccurate about taking an average rate over a large interval. As we take the average rate over a smaller and smaller interval, it becomes closer to describing the behavior of the [...]