12.1 Gradients of curves (PURE 1- Chapter 12: Differentiation)

Summary

This section of Chapter 12 explores the concept of differentiation, specifically focusing on gradients of curves. It revisits the idea of gradients from straight lines, where the slope is constant, and extends it to curves where the slope changes at different points. By using tangents, students can find the gradient at any point on a curve. The chapter demonstrates calculating gradients for a quadratic curve, showing how tangent lines vary. Through examples, it emphasizes that as a point approaches another point, the gradient estimate becomes more accurate, converging toward an exact value.

Takeaways

• 📐 The chapter is about finding the gradient of curves, which differs from straight lines.
• 📊 At GCSE level, gradients are found using the equation y = mx + c, where m is the gradient.
• 📈 For curves, the gradient changes at different points and is found using a tangent line.
• ⚖️ A tangent touches the curve at one point, and the gradient of the tangent gives the gradient at that point on the curve.
• ➖ At certain points, the gradient is negative (when the tangent slopes downwards), and at the bottom of the curve, the gradient is zero (tangent is horizontal).
• 📝 Example 1 shows the curve y = x² and explains how to calculate the gradient of the tangent line at specific points.
• 🔢 Gradients are calculated using the formula (y₂ - y₁) / (x₂ - x₁), as seen in straight-line graphs.
• 🔄 As the point P moves closer to point A, the gradient of the chord approaches the gradient of the tangent, which gets closer to 2.
• 🔍 The final formula for the gradient is derived as 2 + h, where h gets smaller as the point moves closer, converging to the exact gradient.
• 📚 As point P approaches point A, the estimated gradient becomes more accurate, reinforcing that the tangent's gradient at A is exactly 2.

Q & A

What is the definition of the gradient as explained in the video?

The gradient is the slope or steepness of a line or curve. For straight lines, it remains constant, but for curves, it changes at different points.

How is the gradient of a tangent line to a curve calculated?

The gradient of a tangent is found by drawing a line that touches the curve at a specific point, then calculating the gradient of that tangent line.

What happens to the gradient of a curve at different points?

The gradient changes depending on the point on the curve. At some points, it may be positive, negative, or even zero. For example, at the bottom of a curve, the gradient is zero as the tangent line is horizontal.

How is the gradient different between a straight line and a curve?

For a straight line, the gradient is constant, while for a curve, the gradient varies at different points based on the steepness of the curve.

What formula is used to calculate the gradient of a straight line?

The formula for the gradient of a straight line is (y2 - y1) / (x2 - x1), which is derived from the slope formula covered in straight line graph topics.

How does the gradient of the tangent relate to the curve at point A in the example?

In the example, the gradient of the tangent line at point A (1,1) is calculated as 2, which represents the slope of the curve at that specific point.

How does the gradient of the chord change as point P gets closer to point A?

As point P moves closer to point A, the gradient of the chord becomes closer to the gradient of the tangent, approaching a value of 2.

What does the expression '2 + h' represent in the context of gradient calculation?

'2 + h' is the expression used to estimate the gradient as point P moves closer to point A. As h approaches zero, the gradient converges to 2.

Why is the gradient of the tangent considered an exact value while the gradient of the chord is an estimate?

The gradient of the tangent is exact because it is calculated directly at point A, while the gradient of the chord is an estimate as it depends on the distance between points A and P.

What conclusion can be drawn as point P gets infinitely closer to point A?

As point P gets infinitely closer to point A, the gradient of the chord approaches the exact gradient of the tangent, which is 2.