Understanding how to estimate gradients by drawing tangents on curves is a fundamental aspect of calculus that allows us to analyse the behaviour of functions at specific points. This section is dedicated to mastering these techniques, providing a clear pathway to comprehend the rate of change of functions graphically.

Introduction to Gradient Estimation

The gradient of a curve at a given point reflects the slope of the tangent at that point. It's an essential concept, especially when we delve into rates of change in various scientific disciplines.

What is a Tangent?

• A tangent is a straight line that lightly touches a curve at one point, representing the direction in which the curve is heading at that point.

Estimating Gradients: The Process

Estimating a curve's gradient involves drawing a tangent at the point of interest and calculating this tangent's slope using the formula:

Drawing Tangents: A Step-by-Step Guide

1. Identify the Point of Interest: Choose the specific point on the curve where you want to estimate the gradient.

2. Draw the Tangent: Sketch a straight line that just touches the curve at the chosen point.

3. Select Points on the Tangent: Pick two points on the tangent line, ideally far apart, for a more accurate gradient calculation.

4. Calculate the Gradient: Use the chosen points' coordinates to calculate the tangent's slope, providing an estimate of the curve's gradient at your point of interest.

Example: Estimating the Gradient of $y = x^2$ at $x = 2$

Let's estimate the gradient of the curve $y = x^2$ at the point where $x = 2$.

Drawing the Curve and Tangent

1. We first sketch the curve of $y = x^2$.

2. At $x = 2$, we draw a tangent to the curve. Upon drawing the tangent, we select two points on this line, for instance, $x = -3$ and $x = 5$, and calculate the gradient of the tangent. For our example, the calculated gradient at $x = 2$ is 4, and the equation of the tangent line can be described by $y = 4x - 4$.

2. This tangent touches the curve precisely at the point $(2, 4)$. The slope of the tangent, which represents the estimated gradient of the curve at $x = 2$, is 4, and the y-intercept of the tangent line is -4.

Practice Problems

Problem 1:

Estimating the Gradient of $y = 3x^3 - 2x^2 + x - 5$ at $x = 1$.

Solution:

1. Sketch the Curve: Draw the curve $y = 3x^3 - 2x^2 + x - 5$.

2. Tangent at $x = 1$: Draw a tangent at the point $x = 1$, and calculate its slope.

• The derivative of $y$ with respect to $x$ at $x = 1$ gives the slope of the tangent.
• Use the derivative formula to find the exact slope and proceed with the gradient calculation similarly to the example above.

Problem 2:

Determine the Gradient of $y = \sqrt{x} \text{ at } x = 4$

Solution:

1. Graph the Function: Plot $y = \sqrt{x}$ on a graph.

2. Draw and Calculate: At $x = 4$, draw a tangent and estimate its gradient.

• Again, the slope of the tangent at $x = 4$ can be found by deriving $y$ with respect to $x$ and evaluating at $x = 4$.
• Follow the steps to calculate the gradient as demonstrated.