The point P(0.5,0) lies on the curve y=cosπx. (a) If Q is...

The point $P(0.5,0)$ lies on the curve $y=\cos \pi x.$ (a) If $Q$ is the point $(x, \cos \pi x),$ use your calculator to find the slope of the secant line $P Q$ (correct to six decimal places) for the following values of $x$ : $\begin{array}{llll}\text { (i) } 0 & \text { (ii) } 0.4 & \text { (iii) } 0.49 & \text { (iv) } 0.499\end{array}$ $\begin{array}{llll}\text { (v) } 1 & \text { (vi) } 0.6 & \text { (vii) } 0.51 & \text { (viii) } 0.501\end{array}$ (c) Using the slope from part (b), find an equation of the tangent line to the curve at $P(0.5,0).$ (d) Sketch the curve, two of the secant lines, and the tangent line.

The point $P(0.5,0)$ lies on the curve $y=\cos \pi x.$
(a) If $Q$ is the point $(x, \cos \pi x),$ use your calculator to find the slope of the secant line $P Q$ (correct to six decimal places) for the following values of $x$ :
$\begin{array}{llll}\text { (i) } 0 & \text { (ii) } 0.4 & \text { (iii) } 0.49 & \text { (iv) } 0.499\end{array}$
$\begin{array}{llll}\text { (v) } 1 & \text { (vi) } 0.6 & \text { (vii) } 0.51 & \text { (viii) } 0.501\end{array}$
(c) Using the slope from part (b), find an equation of the tangent line to the curve at $P(0.5,0).$
(d) Sketch the curve, two of the secant lines, and the tangent line.

Key Concepts

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 over an interval between these two points, providing a finite approximation to the slope of the curve on that interval.

Tangent Line

A tangent line touches a curve at one point and has the same instantaneous rate of change as the curve at that point. It is the best linear approximation of the curve near the point of tangency, and its slope is given by the derivative of the function at that point.

Slope of a Line

The slope of a line is defined as the ratio of the vertical change to the horizontal change between two points on the line. This concept quantifies how steep the line is and is fundamental in understanding rates of change in calculus.

Difference Quotient

The difference quotient is an expression used to calculate the slope of the secant line between two points on a function. It is formed by the ratio (f(x+h) - f(x)) / h and serves as the basis for defining the derivative as the interval h approaches zero.

Derivative

The derivative of a function at a certain point is the limit of the difference quotient as the interval between the points approaches zero. It represents the instantaneous rate of change of the function and determines the slope of the tangent line to the curve at that point.

Calculator Approximation

Calculator approximation involves using numerical methods and computational tools to estimate values such as slopes of secant lines and derivatives. This approach is useful when exact analytical solutions are difficult to determine or when a practical estimation is sufficient for understanding the behavior of the function.

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