Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. For linear functions, the slope of any secant line always equals the slope of any tangent line. b. The slope of the secant line passing through the points \(P\) and \(Q\) is less than the slope of the tangent line at \(P\). c. Consider the graph of the parabola \(f(x)=x^{2} .\) For \(x > 0\) and \(h > 0,\) the secant line through \((x, f(x))\) and \((x+h, f(x+h))\) always has a greater slope than the tangent line at \((x, f(x))\)

Short answer

Question: Analyze the given statements and determine if they are true or false. If false, provide a counterexample. The statements are: a) For a linear function, the slope of any secant line is equal to the slope of any tangent line. b) The slope of the secant line passing through points P and Q is always less than the slope of the tangent line at point P. c) For the graph of a parabola f(x) = x^2 with x > 0 and h > 0, the slope of the secant line passing through (x, f(x)) and (x+h, f(x+h)) is always greater than the slope of the tangent line at (x, f(x)). Answer: a) True. The slope of any secant line and tangent line for a linear function will always be equal, as the linear function has a constant rate of change. b) False. A counterexample is a line with a positive slope, in which case the slope of the secant line between points P and Q has the same slope as the tangent line at point P. c) True. For the given parabolic function, the slope of the secant line will always be greater than the slope of the tangent line when x > 0 and h > 0.

Step-by-step solution

Statement a: Slope of Secant vs Tangent Lines for Linear Functions

It's given that for a linear function, the slope of any secant line is equal to the slope of any tangent line. A linear function can be written in the form of \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Since a linear function has a constant rate of change, its slope is constant everywhere. Therefore, a secant line between any two points on a linear function will have the same slope as the function itself. A tangent line at any point on the linear function will also have the same slope. Thus, because both slopes are equal, this statement is true.

Statement b: Comparing Slopes of Secant Line vs Tangent Line at P-Q

The statement says that the slope of the secant line passing through points \(P\) and \(Q\) is always less than the slope of the tangent line at point \(P\). There is no specific function mentioned in this scenario, so it's incorrect to say that the slope of the secant line is always less than the slope of the tangent line at point \(P\) in every situation. For instance, consider a line with a positive slope: the secant line between points \(P\) and \(Q\) has the same slope as the tangent line at point \(P\), which contradicts the statement. So, since a counterexample has been found, this statement is false.

Statement c: Slopes of Secant and Tangent Lines on Parabola

The given information is that the graph is of a parabola \(f(x) = x^2\) with \(x > 0\) and \(h > 0\). The secant line passes through \((x, f(x))\) and \((x+h, f(x+h))\), and the tangent line is at \((x, f(x))\). First, we need the derivative of \(f(x) = x^2\), which gives us the slope of the tangent line. The derivative is \(f'(x) = 2x\). So, for \(x > 0\), the slope of the tangent line is \(2x\). For the secant line, the slope can be calculated using the slope formula: \(m_{sec} = \frac{f(x+h) - f(x)}{(x+h)-x} = \frac{(x+h)^2 - x^2}{h} = \frac{x^2 + 2xh + h^2 - x^2}{h} = 2x + h\). From this, it's evident that the slope of the secant line is \(2x + h\). Since \(x > 0\) and \(h > 0\), the secant line's slope will always be greater than the tangent line's slope, making this statement true.

Key concepts

Linear Functions

In the realm of calculus, understanding linear functions is fundamental. These functions are straight lines with the general equation being expressed as \( y = mx + b \), where \( m \) represents the slope, and \( b \) signifies the y-intercept. One of the distinctive characteristics of linear functions is that the slope remains constant, irrespective of which two points on the line you choose to consider.

This constancy of slope means it's the same for secant lines, which intersect the graph at two points, and tangent lines, which merely touch it at one point without intersecting. This feature is crucial for understanding the overall behavior of linear graphs and sets the stage for exploring more complex functions.

Slope of Secant Line

The slope of a secant line demonstrates the average rate of change between two points on a curve. To find the slope of a secant line, one uses the formula \( m_{sec} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \), where \( x_1 \) and \( x_2 \) mark the x-coordinates of the two points, and \( f(x_1) \) and \( f(x_2) \) their respective y-values.

However, it's a misconception to believe that the secant line's slope is always less than or always greater than the slope of the tangent line at one of its endpoints. The comparison varies depending on the function's specific shape and the interval chosen.

Slope of Tangent Line

Diving deeper into calculus brings us to the concept of the slope of a tangent line. This slope represents the instantaneous rate of change of the function at a specific point. It answers the question: 'At this exact point, how steep is the curve?' Unlike the secant line, which considers two points, the tangent line's slope is derived just at one point, making it a powerful tool in differential calculus.

Calculating this slope requires the derivative of the function at the point of interest. For instance, if the function is \( y = f(x) \), then the slope of the tangent line at \( x \) is \( f'(x) \) where \( f' \) denotes the derivative of \( f \). The closer a secant line's two points get to each other, the more the secant line resembles the tangent line, illuminating the concept of the derivative as the limit of the slopes of secant lines.

Parabolic Functions

Parabolic functions, which are of the form \( f(x) = ax^2 + bx + c \), represent curves that are symmetric and open upwards or downwards, contingent on the sign of \( a \). In particular, when \( a \) is positive, the parabola opens upward, as seen with the standard \( f(x) = x^2 \).

An intriguing property of parabolas is that the slope of their tangent lines continuously change as one moves along the curve. Secant lines drawn between points on the parabola can have a variety of slopes, typically showcasing how the curve's steepness increases or decreases between those points. As with linear functions, understanding how the slope behaves on parabolic functions deepens one's comprehension of the nature of these curves.

Derivatives

Derivatives stand at the core of calculus, symbolizing the heartbeat of change. They quantify how a function changes at any given point, offering a precise mathematical description of rates of change and slopes of tangent lines. Essentially, \( f'(x) \) is the derivative of a function \( f(x) \), indicating the slope of the curve at point \( x \).

Derivatives are powerful because they can predict behavior; for example, knowing the derivative allows one to understand whether a function is increasing or decreasing, and at what rate, at any point along the graph. They articulate the dynamic nature of functions and hence, are pivotal in mathematics, physics, engineering, economics, and beyond.