A tangent line shows the instantaneous direction of a curve at one specific point. In calculus, this direction is measured by the derivative, which gives the slope of the curve at that point. Tangent lines matter because they let us approximate curved behavior with a straight line near a point.
Normal lines are equally important because they show the direction perpendicular to the tangent.
Key Facts
- The slope of the tangent line to y = f(x) at x = a is m_t = f'(a).
- The point of tangency is (a, f(a)).
- Point-slope form for the tangent line is y - f(a) = f'(a)(x - a).
- If the tangent slope is m_t, the normal slope is m_n = -1/m_t when m_t is not 0.
- Point-slope form for the normal line is y - f(a) = m_n(x - a).
- A tangent line with slope 0 has a vertical normal line x = a.
Vocabulary
- Tangent line
- A tangent line is the line that touches a curve at a point and has the same slope as the curve there.
- Normal line
- A normal line is the line through the point of tangency that is perpendicular to the tangent line.
- Derivative
- The derivative f'(x) gives the instantaneous rate of change or slope of a function at each x-value.
- Point of tangency
- The point of tangency is the point where the tangent line meets the curve and matches its local direction.
- Point-slope form
- Point-slope form is the line equation y - y1 = m(x - x1), using a known point and slope.
Common Mistakes to Avoid
- Using f(a) as the slope instead of f'(a) is wrong because f(a) gives the y-value, while f'(a) gives the tangent slope.
- Forgetting to evaluate the derivative at x = a is wrong because f'(x) is a slope function, not the final numerical slope at the chosen point.
- Using the same slope for the normal line is wrong because the normal line must be perpendicular to the tangent, so its slope is the negative reciprocal when the tangent slope is nonzero.
- Writing the line equation with the wrong point is wrong because both tangent and normal lines must pass through (a, f(a)), not just any point near the curve.
Practice Questions
- 1 Find the tangent and normal lines to f(x) = x^2 + 1 at x = 2.
- 2 For f(x) = 3x^3 - x, find the equation of the tangent line at x = 1.
- 3 A curve has a horizontal tangent at point (4, 7). Describe the normal line at that point and explain why it has that form.