Gradients, Tangents & Normals

• Gradient of a Curve: Unlike a straight line, the gradient of a curve changes at every point. The derivative, $f'(x)$ or $\frac{dy}{dx}$, provides a general formula for this changing gradient.

• The Tangent: A tangent to a curve at a specific point is a straight line that just touches the curve at that point and has the exact same gradient as the curve at that coordinate.

• The Normal: A normal to a curve at a specific point is a straight line that passes through that point and is perpendicular (at a 90-degree angle) to the tangent line.

• Perpendicularity Rule: Two lines are perpendicular if the product of their gradients is $-1$. If the tangent has gradient $m$, the normal has gradient $-\frac{1}{m}$.

• The Derivative as a Slope Function: The fundamental principle of calculus is that $f'(x)$ represents the instantaneous rate of change. Geometrically, this is the slope of the tangent line at any point $x$.

• Linearity of Tangents and Normals: Because tangents and normals are straight lines, they must satisfy the linear equation $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the point of contact.

• Negative Reciprocals: The relationship between perpendicular gradients ($m_1 m_2 = -1$) is a geometric constant. This ensures that the normal always meets the tangent at a right angle, regardless of the curve's complexity.

Step-by-Step: Finding the Equation

• Step 1: Differentiate: Find the first derivative $f'(x)$ of the given function $y = f(x)$. This provides the gradient function.

• Step 2: Find the Gradient ($m$): Substitute the specific $x$-coordinate (let's call it $a$) into the derivative: $m = f'(a)$.

• Step 3: Find the $y$-coordinate: If only the $x$-coordinate is given, substitute $x = a$ into the original function $f(x)$ to find the corresponding $y$-value.

• Step 4: Select the Gradient: For a Tangent, use $m$. For a Normal, use the perpendicular gradient $m_{normal} = -\frac{1}{m}$.

• Step 5: Construct the Equation: Use the point-slope formula: $y - f(a) = m(x - a)$

• It is vital to distinguish between the function itself and its derivative when performing substitutions.

• The 'Original Function' Error: A common mistake is substituting the $x$-coordinate into the original function $f(x)$ to find the gradient. Remember: $f(x)$ gives the height ($y$), while $f'(x)$ gives the slope ($m$).

• The 'Reciprocal' Slip: Students often forget the negative sign when finding the normal gradient, using $\frac{1}{m}$ instead of $-\frac{1}{m}$.

• Algebraic Rearrangement: Ensure you can rearrange the final equation into the requested format, such as $y = mx + c$ or $ax + by + c = 0$.