This picture (the after picture) shows how the Leaning Tower of Pisa has been completely straightened. Click on it and magnify it to see the beauty of the structure.
Here's a proof( in fact, two proofs) that a secant line lies above the curve of a concave up function.We also show that slopes of secant lines are an increasing function of x.We had shown in class that concave up arcs are above any tangent on that arc.
This file illustrates ( but does not prove) that the perpendicular bisectors of the sides of a triangle are concurrent i.e., they meet in the same point . It also illustrates how we can define generic linear functions given two points on the line or one point and the slope.
The file shows that a differentiable function must be continuous and expresses continuity in various forms recognizable as similar to numerators of the various difference quotients that we have used to define derivatives.We also see how maple can compute derivatives.
This lab involves some applications of integration such as finding the mean value of a function over an interval, finding the total distance traveled in one dimensional motion for s specified time interval given the velocity function, and finding areas using vertical, horizontal strips or just some geometry.
Practice plotting any horizontal and inflectional tangents to a cubic f(x)that you will define.Also, verify that your cubic is symmetric to its inflection point i.e.,if f ''(a) =0 then f(a)=(1/2)(f(a-u)+f(a+u))
Here we use maple to find the absolute extrem of a continuous function on closed interval and we get a preview of the Mean Value Theorem .You will also (If time permits) have maple determine the sign of first and second derivatives.We will later see how these signs are reflected in graphs.
Here we investigate the "mean value point" for y =x^2 and y=x^3.An optional template is also provided showing a "display" option for graphs.We define the mean value point as the point on the arc of the curve between (a,f(a)) and (b,(f(b)) where the tangent is parallel to the secant
This document shows two different plots of position,velocity and acceleration plots.Speed increases when velocity and acceleration have the same sign.This should be most clear in the second plot.
