Problem 4 - Calculus - AP Edition

Question

Given that the tangent line to \(y=f(x)\) at the point (1,2) passes through the point \((-1,-1),\) find \(f^{\prime}(1)\)

Step-by-Step Solution

Answer

\(f'(1) = \frac{3}{2}\)

Step 1: Determine Slope of the Tangent Line

To find the slope of the line that passes through two points, we use the formula for the slope, given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, the points are \((1, 2)\) and \((-1, -1)\). So, the slope \( m = \frac{2 - (-1)}{1 - (-1)} = \frac{3}{2} \). Thus, the slope of the tangent line at \((1, 2)\) is \(\frac{3}{2}\).

Step 2: Relate Slope to Derivative

The derivative of a function at a point gives the slope of the tangent line at that point. Since we found the slope of the tangent line at \( (1, 2) \) is \( \frac{3}{2} \), it follows that the derivative of \(f(x)\) at \( x = 1 \), which is \( f'(1) \), must also be \( \frac{3}{2} \).

Key Concepts

Tangent Line

A tangent line is a straight line that just touches a curve at a given point. This means it only meets the curve at one spot and isn't crossing it. Imagine tracing a curve with a pencil; the tangent line would just skim a single dot on the curve without cutting through.

In calculus, the tangent line is especially important because it reflects the direction in which the curve is heading at that exact point. It acts like a snapshot of the curve's trend at one specific spot.

• Tangent lines are useful in understanding the local behavior of functions.
• They provide insights into how a curve bends and are crucial in many calculations.
• In geometry, a tangent line to a circle is always perpendicular to the radius drawn to the point of tangency.

Recognizing the tangent line can greatly help in interpreting and predicting the behavior of graphs and shapes.

Slope

The slope is a key concept when discussing lines and curves. It describes how steep or flat a line is and shows the rate at which one quantity changes with respect to another. When looking at a line on a graph, the slope can be seen as "rise over run," representing the change in

• "Rise" – how much the line goes up or down as you move along.
• "Run" – how far you go horizontally on the x-axis for a particular change on the y-axis.

Mathematically, it's calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where

• \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line.

For example, with points \((1, 2)\) and \((-1, -1)\), the slope is \(\frac{3}{2}\), telling us the line rises 3 units for every 2 units it runs to the right. A higher slope means a steeper line, while a slope of zero indicates a flat line.

Slope is critical in calculus because it's tied directly to the concept of a derivative, which represents the slope of the tangent line at any given point on a curve.

Calculus

Calculus is a branch of mathematics focused on change, motion, and the rates at which quantities differ from one another. It's often divided into two main areas: differential calculus and integral calculus. In the context of our exercise, we're dealing with differential calculus.

Differential calculus is largely concerned with finding derivatives. Derivatives represent the instantaneous rate of change of a function, giving us the slope of the tangent line at any given point.

Here's why derivatives and calculus are vital:

• They help us understand and predict the behavior of functions and systems in motion.
• They are essential in diverse fields like physics, engineering, economics, and beyond, whenever change over time is involved.
• Derivatives can reveal acceleration, growth rates, and help optimize real-world scenarios.

By mastering the core principles of calculus, problems such as finding the slope of a tangent line become more approachable and understandable. Calculus allows us to delve deeper into the intricacies of mathematical functions, providing the insight necessary for advanced study and application.