Chapter 2 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 2.7: Derivatives at endpoints are one-sided limits.
Figure 2.9: We made the graph of y´ = ƒ´(x) in (b) by plotting slopes from the graph of y = f (x) in (a). The vertical coordinate of B´ is the slope at B and so on. The graph of y´ = f ´(x) is a visual record of how the slope of f changes with x.
Figure 2.16: The velocity graph for Example 3.
Figure 2. 18: (a) The rock in Example 5 Figure 2.18: (a) The rock in Example 5. (b) The graphs of s and v as functions of time; s is largest when v = ds/dt = 0. The graph of s is not the path of the rock: It is a plot of height versus time. The slope of the plot is the rock’s velocity graphed here as a straight line.
Figure 2.25: The curve y´ = –sin x as the graph of the slopes of the tangents to the curve y = cos x.
Figure 2.28: When gear A makes x turns, gear B makes u turns and gear C makes y turns. By comparing circumferences or counting teeth, we see that y = u/2 and u = 3x, so y = 3x/2. Thus, dy/du = 1/2, du/dx = 3, and dy/dx = 3/2 = (dy/du)(du/dx).
Figure 2.31: sin (x°) oscillates only /180 times as often as sin x oscillates. Its maximum slope is /180. (Example 9)
Figure 2. 39: The graph of y2 = x2 + sin xy in Example 2 Figure 2.39: The graph of y2 = x2 + sin xy in Example 2. The example shows how to find slopes on this implicitly defined curve.
Figure 2.40: Example 3 shows how to find equations for the tangent and normal to the curve at (2, 4).
Figure 2.43: The balloon in Example 3.
Figure 2.44: Figure for Example 4.
Figure 2.45: The conical tank in Example 5.
Figure 2.48: The graphs of inverse functions have reciprocal slopes at corresponding points.
Figure 2.49: The graph of y = sin–1 x has vertical tangents at x = –1 and x = 1.
Figure 2.51: The position of the curve y = (a h – 1) /h, a > 0, varies continuously with a.
Figure 2.52: The tangent line intersects the curve at some point (a, ln a,) where the slope of the curve is l/a. (Example 4)