Tangent line parametric curve equations

For a parametric curve defined by x(t) and y(t), the slope of the tangent line is dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0. This ratio of derivatives is the direct counterpart to the slope formula for explicit functions. The most common error is dividing by zero: when dx/dt = 0, the tangent is vertical, not horizontal. You must separately check dy/dt = 0 for horizontal tangents. This method comes from OpenStax Calculus Vol. 2 section 7.2, which covers the calculus of parametric curves.

To compute the tangent line at a specific t-value: find the point (x(t), y(t)), then evaluate dy/dt and dx/dt at that t, and write the line in point-slope form: y, y₁ = m (x, x₁). If the denominator is zero, the line is x = x₁. If both derivatives are zero, the curve may have a cusp or a self-intersection, the tangent is not defined at that point.

Example: x = t², y = t³, 3t at t = 2.

First, find the point: x(2) = 4, y(2) = 8-6 = 2, so (4, 2). Next, compute derivatives: dx/dt = 2t, so at t=2, dx/dt = 4. dy/dt = 3t², 3, so at t=2, dy/dt = 12-3 = 9. The slope m = (9)/(4) = 2.25. The tangent line in point-slope form: y, 2 = 2.25 (x, 4).

Failure case: If t=0 for both x and y, say x = t², y = t³, then dx/dt = 0 and dy/dt = 0 at t=0. The slope is indeterminate (0/0), and the curve has a cusp at (0,0). The tangent line does not exist in the usual sense. Check differentiability before applying the formula.

For a parametric curve, horizontal tangents occur where dy/dt = 0 and dx/dt ≠ 0. At those points, the slope is zero. Vertical tangents occur where dx/dt = 0 and dy/dt ≠ 0. At those points, the slope is undefined, the tangent line is x = constant.

Example: For x = t², 4, y = t³, 3t, find horizontal tangents. Set dy/dt = 3t², 3 = 0 → t = ±1. At t=1, dx/dt = 2, so slope = 0. Point: x =, 3, y =, 2. Tangent: y =, 2. At t=, 1, dx/dt =, 2, slope = 0. Point: x =, 3, y = 2. Tangent: y = 2. For vertical tangents, set dx/dt = 2t = 0 → t=0. Then dy/dt =, 3, so denominator zero. Point: x=, 4, y=0. Tangent: x =, 4.

This distinction matters for curve sketching and for AP Calculus free-response questions. The OpenStax Calculus Vol. 2 section 7.2 condition is: dx/dt ≠ 0 for a finite slope; dy/dt = 0 and dx/dt ≠ 0 for a horizontal tangent; dx/dt = 0 and dy/dt ≠ 0 for a vertical tangent.

For a polar curve r = f(θ), you cannot directly apply the parametric slope formula to r and θ. You must first convert to parametric form using x = r cosθ, y = r sinθ. Then the slope dy/dx becomes:

dy/dx = (r' sinθ + r cosθ) / (r' cosθ, r sinθ), where r' = dr/dθ.

This formula comes from OpenStax Calculus Vol. 2 section 7.4, which covers polar area, arc length, and tangent slopes. The numerator gives the condition for horizontal tangents (set = 0, denominator ≠ 0); the denominator gives the condition for vertical tangents (set = 0, numerator ≠ 0). At the pole (r=0), the tangent slope simplifies to tanθ, provided dr/dθ ≠ 0.

Example: Find the tangent line to r = 1 + cosθ at θ = π/2.

First, compute r(π/2) = 1 + 0 = 1. Then r' =, sinθ, so at π/2, r' =, 1. The parametric point: x = r cosθ = 1·0 = 0; y = r sinθ = 1·1 = 1. So the point is (0, 1).

Now compute the slope: numerator = r' sinθ + r cosθ = (, 1)(1) + (1)(0) =, 1. Denominator = r' cosθ, r sinθ = (, 1)(0), (1)(1) =, 1. So m = (, 1)/(, 1) = 1. The tangent line: y, 1 = 1(x, 0), or y = x + 1.

Failure case: At θ = π, r = 1 + (, 1) = 0. The pole. The slope is tan(π) = 0, so the tangent is horizontal (y=0). But if dr/dθ = 0 at the pole, the tangent direction is undefined, the curve may have a cusp at the origin.

For parametric curves, the second derivative d²y/dx² gives concavity. Compute it as (d/dt (dy/dx)) / (dx/dt). This tells you whether the curve is concave up (positive) or concave down (negative) at a point. For polar curves, convert to parametric first, then use the same formula. The second derivative is not needed for the tangent line itself, but it is necessary for curve sketching and for understanding the behavior near the point of tangency, for instance, whether the curve crosses the tangent line (inflection point) or stays on one side.

For the AP Calculus BC exam, you may be asked to find d²y/dx² for a parametric curve and use it to classify concavity. The OpenStax Calculus Vol. 2 section 7.2 covers this derivation.

Parametric slope: dy/dx = (dy/dt)/(dx/dt), dx/dt ≠ 0.

Horizontal tangent: dy/dt = 0, dx/dt ≠ 0.

Vertical tangent: dx/dt = 0, dy/dt ≠ 0.

Polar slope (converted): dy/dx = (r' sinθ + r cosθ)/(r' cosθ, r sinθ).

Polar horizontal: r' sinθ + r cosθ = 0, denominator ≠ 0.

Polar vertical: r' cosθ, r sinθ = 0, numerator ≠ 0.

Pole tangent: slope = tanθ, provided dr/dθ ≠ 0.

Example 1 (Parametric): x = t², y = t³, at t=2. Slope = 3, line y - 8 = 3(x - 4).

Example 2 (Parametric vertical): x = t² - 4, y = t³ - 3t at t=0. Slope undefined (dx/dt=0), line x = -4.

Example 3 (Polar):

This material suits AP Calculus BC students who need to solve parametric and polar tangent problems for the exam, college Calculus 2 students working through curve sketching and linear approximations, and tutors explaining why the 'touches without crossing' definition fails. Self-studying learners with a solid grasp of derivatives can use these steps to check their analytic work.

If you do not know what a derivative is or how to compute basic limits, study the limit definition of the derivative and the power rule first (OpenStax Calculus Vol. 1 sections 2.1, 3.1, 3.3). You can differentiate polynomial, trigonometric, exponential, and logarithmic functions. The single thing that most often goes wrong: forgetting to check that dx/dt ≠ 0 before applying the parametric slope formula, or failing to convert polar to parametric before computing dy/dx. Do that check every time.