The point P(1, (1)/(2)) lies on the curve y=x /(1+x). (a) If...

The point $P\left(1, \frac{1}{2}\right)$ lies on the curve $y=x /(1+x).$ (a) If $Q$ is the point $(x, x /(1+x))$, use your calculator to find the slope of the secant line $P Q$ (correct to six decimal places) for the following values of $x$ : $\begin{array}{llll}\text { (i) } 0.5 & \text { (ii) } 0.9 & \text { (iii) } 0.99 & \text { (iv) } 0.999\end{array}$ $\begin{array}{llll}\text { (v) } 1.5 & \text { (vi) } 1.1 & \text { (vii) } 1.01 & \text { (viii) } 1.001\end{array}$ (b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at $P\left(1, \frac{1}{2}\right).$ (c) Using the slope from part (b), find an equation of the tangent line to the curve at $P\left(1, \frac{1}{2}\right).$

The point $P\left(1, \frac{1}{2}\right)$ lies on the curve $y=x /(1+x).$
(a) If $Q$ is the point $(x, x /(1+x))$, use your calculator to find the slope of the secant line $P Q$ (correct to six decimal places) for the following values of $x$ :
$\begin{array}{llll}\text { (i) } 0.5 & \text { (ii) } 0.9 & \text { (iii) } 0.99 & \text { (iv) } 0.999\end{array}$
$\begin{array}{llll}\text { (v) } 1.5 & \text { (vi) } 1.1 & \text { (vii) } 1.01 & \text { (viii) } 1.001\end{array}$
(b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at $P\left(1, \frac{1}{2}\right).$
(c) Using the slope from part (b), find an equation of the tangent line to the curve at $P\left(1, \frac{1}{2}\right).$

Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function, which describes how the function changes at any point. This involves applying limit concepts to compute the slope of the tangent line for the curve. Differentiation is a central tool in calculus, used for analyzing dynamic systems and solving problems involving rates of change.

Equation of a Line

The equation of a line can be determined if a point on the line and its slope are known. It is typically expressed in the point-slope form, which facilitates the construction of the line's equation once the slope (from the derivative for tangent lines) is calculated. This concept is key in linking derivative calculations to geometric representations of functions.

Limit Process

The limit process involves analyzing the behavior of a function as the input approaches a particular value. In the context of derivatives, it is used to determine the instantaneous rate of change by taking the limit of the difference quotient as the interval between the two points approaches zero. This process provides a rigorous means of defining and calculating derivatives.

Tangent Line

A tangent line to a curve at a point is a straight line that touches the curve at that point without crossing it. It represents the instantaneous rate of change or the derivative of the function at that point. Tangent lines are fundamental in differentiating functions and analyzing their behavior near specific points.

Secant Line

A secant line is a line that intersects a curve at two distinct points. It represents the average rate of change of the function between those two points. In the context of calculus, calculating the slope of a secant line provides an approximation for the derivative, especially when the two points are close together on the curve.

Difference Quotient

The difference quotient is an expression that calculates the average rate of change of a function over an interval. It is defined as the ratio of the change in the function's value to the change in the input value between two points. This concept is essential as it forms the basis for defining the derivative through a limiting process.

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