Calculus

• How does the gradient of the curve change?


  The gradient changes depending on the position along the curve.
• What does the gradient of the tangent at a point on the curve represent?

  It represents the steepness of the curve at that point.
• How can you approximate the gradient at a point on the curve?


  By calculating the gradient of a chord between two points on the curve.
• What happens to the gradient of the chord as the points get closer together?


  The gradient of the chord approaches the gradient of the tangent.
• What is the gradient of the chord between the points (2, 4) and (3, 9)?


  The gradient is 5.
• What is the gradient of the chord between the points (2, 4) and (2.5, 6.25)?


  The gradient is 4.5.
• What is the gradient of the chord between the points (2, 4) and (2.01, 4.41)?


  The gradient is 4.01.
• What is the formula for the gradient of a chord?


  The formula is \(\frac{Y_2 - Y_1}{X_2 - X_1}\).
• What does the limit of the gradient approach as the distance \(H\) approaches zero?


  The limit approaches \(2x\) for the curve \(y = x^2\).
• What is the result of differentiating \(y = x^2\)?


  The result is \(\frac{dy}{dx} = 2x\).
• What is the result of differentiating \(y = x^3\)?


  The result is \(\frac{dy}{dx} = 3x^2\).
• What is the general rule for differentiating \(y = x^n\)?


  The result is \(\frac{dy}{dx} = nx^{n-1}\).
• What is the process of going from the curve to its gradient called?

  The process is called differentiation.
• What notation is commonly used for differentiation?


  The notation is \(\frac{dy}{dx}\).
• What are the steps to differentiate \(y = x^2\)?


  1. Identify the function: \(y = x^2\).
  2. Apply the power rule: bring down the exponent.
  3. Reduce the exponent by one.
  4. Result: \(\frac{dy}{dx} = 2x\).
• What are the steps to differentiate \(y = x^3\)?


  1. Identify the function: \(y = x^3\).
  2. Apply the power rule: bring down the exponent.
  3. Reduce the exponent by one.
  4. Result: \(\frac{dy}{dx} = 3x^2\).
• What is the process of differentiation for polynomial functions?


  • Identify the power of \(x\).
  • Apply the power rule: multiply by the exponent.
  • Decrease the exponent by one.
  • Repeat for each term in the polynomial.
• What is the significance of the limit as \(H\) approaches zero in differentiation?


  • It allows for the precise calculation of the gradient at a specific point.
  • It ensures that the approximation of the gradient becomes exact.
  • It leads to the derivative function.