Slope of a tangent line

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 ... 

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.

For a product, the slope of a tangent line of a graph of concentration vs. time... 

Rate of Reaction Expressed as the Slope of a Tangent Line... 

An instantaneous rate of reaction is the rate of a reaction at some precise point in the reaction. It is obtained from the slope of a tangent line to a concentration-time graph. 

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... 

The first version of Equation (17) suggests that a plot of In cCMC versus 1/Tis a straight line of slope A// ,c/[/ (l + m/n)] if AH°mtc and m/n are independent of T. Backtracking to Equation (9) or (15) shows that n must also be constant with respect to temperature for this to be valid. In fact, n increases with temperature for polyoxyethylene nonionics any temperature dependence of n is generally assumed to be absent in ionic systems. Even if the In cCMC versus /T plot is nonlinear, the second form of Equation (17) allows a value of AH°mc to be evaluated from the slope of a tangent to a plot of In cCMC versus T at a particular temperature. Once AG°ic and AH°mjc are known, the entropy of micellization is readily obtained from AG = AH — TAS. Example 8.4 illustrates the use of these relationships. 

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. 

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 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). 

The formula for each estimate (xjH-i) in terms of the previous estimate (x t) may easily be derived. The graphical representation of one step of the procedure is shown on the next page. The slope of the tangent line is df/dx) = however, two known points on this line are (x/t+i, 0) and (a, / ) so that the slope is also equal to (0 - / )/(a +i — x ). Equating these two expressions for the slope yields... 

The derivative of a hinction y(x) at a point is equivalent to the slope of the tangent line to the graph of the function at that point and is defined as (Fig. 2-66)... 

Now, when some quantity is plotted versus time, the slope of the line tells you how fast that quantity is changing with time. So the slopes of the three curves in Figure 3 measure the rates of change of each concentration. The slope of a curve at a particular point is just the slope of a straight line drawn as a tangent to the curve at that point. Because oxygen is a product and its coefficient in the equation is 1, the slope of the O2 curve is simply the reaction rate. 

The derivative gives the slope of the tangent line to f(x) at each point x and can be used to approximate the response to a small perturbation Ax in the independent variable x ... 

Figure 16-3 Plot of H2 concentration versus time for the reaction of 1.000 M H2 with 2.000 M ICl. The instantaneous rate of reaction at any time, t, equals the negative of the slope of the tangent to this curve at time t. The initial rate of the reaction is equal to the negative of the initial slope (t = 0). The determination of the instantaneous rate at t = 2 seconds is illustrated. (If you do not recall how to find the slope of a straight line, refer to Figure 16-5.)...

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