Week 1 - 1.4, 1.5, 1.6 tangent line concept.pdf - Basic...

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PSU Basic Concepts of Calculus
PSU Calculus is the mathematics of change.Ex: velocities, accelerations, tangent lines, slopes, areas, volumes, arc lengths, centroids, curvatures, and a variety of other concepts that have enabled scientists, engineers, and economists to model real-life situations.Pre-Calculus mathematics is more static.Calculus is more dynamic.Calculus
PSU So, one way to answer the question “What is calculus?” is to say that calculus is a “limit machine” that involves three stages. The first stage is precalculus mathematics, such as the slope of a line or the area of a rectangle. The second stage is the limit process, and the third stage is a new calculus formulation, such as a derivativeor integral.Stages of Calculus
PSU Pre-Calculus vs. Calculus
PSU cont’dPre-Calculus vs. Calculus
PSU Pre-Calculus vs. Calculuscont’d
PSU Pre-Calculus vs. Calculuscont’d
PSU Chapter 11.4 The Tangent and Velocity ProblemIn this section, we will learn:How limits arise when we attempt to find the tangent to a curve or the velocity of an object.
PSU THE TANGENT PROBLEMFor a circle, we could simply follow Euclidand say that a tangent is a line that intersects the circle once and only once.For more complicated curves, that definitionis inadequate.
PSU THE TANGENT PROBLEMThe figure displays two lines land tpassing through a point Pon a curve.The line lintersects only once, but it certainly does not look like what is thought of as a tangent.In contrast, the line tlooks like a tangent, but it intersects twice.Linetis a tangentline to the curve Cat the point PThe word tangentis derived from the Latin word tangents, which means ‘touching.’ Thus, a tangent to a curve is a line that touchesthe curve.In other words, a tangent line should have the same directionas the curve at the point of contact.
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PSU We can let Q approach P.As Qapproaches Palongthe parabola, the correspondingsecant lines rotate about Pand approach the tangent line t.Example 1

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