Tangent line slope

Geometrically, the derivative of y = f(x) at any value is the slope of a tangent line T intersecting the curve at the point P(x,y). Two conditions applying to differentiation (the process of determining the derivatives of a function) are ... 

Secant modulus The secant modulus is the ratio of stress to the corresponding strain at any specific point on the stress-strain curve. As shown in Fig. 2-2(a), the secant modulus is the slope of the line joining the origin and a selected point C on the stress-strain curve this could represent a vertical line at the usual 1 % strain. The secant modulus line is plotted from the initial tangent modulus and where it intersects the stress-strain curve. The plotted line location is also based on the angle used in relation to the initial tangent line from the ab-... 

Figure A 1.2 Determination of instantaneous velocity at various points in a reaction progress curve, from the slope of a tangent line drawn to a specific time point.

We can confirm that on a plot of the mole number nqtz for quartz versus time (Fig. 16.1), this value is the slope of the tangent line and hence the dissolution rate —dwqtz/dt we expect. 

A (a) The 2400-s tangent line intersects the 1200-s vertical line at 0.75 M and reaches 0 M at 3500 s. The slope of that tangent line is thus... 

Exothermic events, such as crystallization processes (or recrystallization processes) are characterized by their enthalpies of crystallization (AHc). This is depicted as the integrated area bounded by the interpolated baseline and the intersections with the curve. The onset is calculated as the intersection between the baseline and a tangent line drawn on the front slope of the curve. Endothermic events, such as the melting transition in Fig. 4.9, are characterized by their enthalpies of fusion (AHj), and are integrated in a similar manner as an exothermic event. The result is expressed as an enthalpy value (AH) with units of J/g and is the physical expression of the crystal lattice energy needed to break down the unit cell forming the crystal. 

The slope of the tangent (AM/At) is the instantaneous rate of the reaction at this time. To determine the rate at a different time, we would need to draw another tangent line. In most kinetic studies, we wish to know the initial rate. The initial rate comes from a tangent drawn to the curve at the very beginning of the reaction. 

The instantaneous rate of a reaction is the rate of the reaction at a particular time. To find the instantaneous rate of a reaction using a concentration-time graph, draw a tangent line to the curve and find the slope of the tangent. A tangent line is like a secant line, but it touches the curve at only one point. It does not intersect the curve. 

While in the first step the deviation from the exact solution stems only from approximating the solution curve by its tangent line, in further steps we calculate the slope at the current approximation y1 instead of the unknown true value (), thereby introducing additional errors. The solution of (5.2) is given by (5.3), and the total error Ei = y(t ) - y for this simple equation is... 

The formulas (5.7) and (5.11) of explixit and implicit Euler methods, respectively, are unsymmetrical, using derivative information only at one end of the time interval of interest. Averaging the slopes of the two tangent lines means using more information, and gives... 

Figure 3.15 Plot of integral heat of solution Aifsoln(n) versus n (= moles H20/moles acid), showing the infinite-dilution limit A/fsoln(oo), the heat of dilution AHdn(ti, n2) from nx to n2, and the differential heat of solution (slope of tangent line) 8H(n ), 8H(n2) for representative concentrations...

Second, plot RsD against p and determine slopes of tangent lines at pressures of interest. Only the calculation at 2100 psig will be shown. 

In this work we shall consider continuous functions of a single real variable. We will take the concept of continuity of a function in the simple geometrical sense, namely, that we can draw the graph of the function without lifting up the pencil. Likewise, we shall say that a function is differentiable wherever the slope of its tangent line is well defined. 

Consider a real function y = f(x) of a real variable x. By this we mean a mapping of the real number x to a unique real number y, given by the rule /. Furthermore, let us assume it to be continuous. We will introduce the concept of the derivative of f(x) with respect to x in terms of the slope of the tangent line at the point x,f(x)). In order to do this, we need to consider three simple constructive rules using the slope of a straight line as our starting point, and Leibniz rule as our keystone. The slope is calculated as a ratio of two displacements rise over run . Hence, we define the derivative of y with respect to r as a quotient of the two corresponding differentials, denoted by dy (the rise ) and dx (the run ) ... 

This rule is the core of differential calculus. We will actually prove that it allows us to interpret the derivative of a function at a point (x, y) as the slope of the tangent line touching it. We shall henceforth refer to the latter simply as the slope of the curve. 

The key point here is to realize that Eq. (7) represents the slope of the parabola at point x. In order to this, we first need to know how the tangent line can be defined in algebraic terms. 

The left-hand side of this equation is the slope of eie, while the right-hand side represents the same function rotated by 90 degrees, so the tangent line turns out to be perpendicular to the radius vector, therefore forming a circle in the complex plane. Furthermore, since the modulus of e e is always unity, the corresponding circle is the unit circle. Euler s formula then asserts that ... 

We have identified the derivative of a function with its slope, or more properly, the slope of its tangent line through various examples. It is now time to prove this statement in general. 

Theorem The slope of the tangent line of a function f(x) at a point x0 is given by its derivative at that point. 

The varying behavior of the multiple steady states in Figure 3.2 is called bifurcation. The bifurcation points for the parameter s are determined by the tangent lines with extreme slopes s and s as depicted by the dashed and dotted lines in Figure 3.2. For any s < s < s there are three steady states, while for any s > s or for any s < s there is only one steady-state solution of the system. And of course, for s = s and s = s there are precisely two steady states. 

It follows from equation (7.8) that unstable crack growth occurs if the slope of the straight line corresponding to the stress intensity factor at constant applied stress is greater than or equal to the slope of the tangent line to the fracture resistance curve at the same point (Fig. 7.6). Also the applied stress intensity factor becomes higher than the fracture resistance of the material. 

We can draw a unique tangent line (a straight line whose slope matches the curve s slope) at each point on the curve. Recall that the slope of a line is defined as the amount y changes if x is changed by one for example, the line y = 3x + 6 has a slope of three. 

Three of these tangent lines are drawn on the curve in Figure 2.1. You can see qualitatively how to draw them, but you cannot tell by inspection exactly what slope to use. This slope can be found by looking at two points Xo and xq + Ax, where Ax (the separation between the two points, pronounced delta x ) is small. We then determine the amount Ay = f(xr> + Ax ) — f(x) that the height of the curve changes between those two points. The ratio Ay/ Ax approaches the slope of the tangent line, in the limit that Ax is very small, and is called the derivative dy/dx. Another common shorthand is to write the derivative of /(x) as fix). 

The d in dy and in dx means A in the limit of infinitesimally small changes. The x=x0 in Equation 2.1 just means evaluated at the pointx = xo The restriction Ax 0 is very important the expression in Equation 2.1 will only give the slope of the tangent line in that limit. You can see from the illustration in Figure 2.1 that a line drawn through the two points xo and (xo + Ax) would be close to the tangent curve, but not on top of it, because Ax is not arbitrarily small. 

FIGURE 2.1 Graph of an arbitrary function f (x). The dashed lines show tangent curves at several points. The slope of the tangent line (called the derivative) can be found by drawing a line between two very close points (here x0 and x0 + Ax). 

Solution The Henry s law constants are found by drawing the tangent lines to the vapor-pressure curves and measuring their slopes. Because these lines begin at the origins (x, = 0 and P, = 0), the slopes are just the intercept with the x, = 1 axis, or from Fig. 2, KAc =150 torr and /Cchi = 140 torr. The accuracy of this procedure is limited by how well the initial slopes can be determined from limited data. (See Problem 6.)... 

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