calc with parametric eqautions

What is a parametric equation?

A way of describing a curve by expressing both x and y as separate functions of a third variable t (the parameter). Instead of y = f(x), you write x = x(t) and y = y(t).

What is the parameter t?

A: The independent variable that drives both x and y. It often represents time. As t increases, the point (x(t), y(t)) moves through the plane and traces the parametric curve.

What is a parametric curve?

The set of all points { (x(t), y(t)) } traced out as t moves through its range [a, b]. It is the path of the moving point — the shape you see when you plot it.

What is the velocity vector in 2D parametric motion? | t

A: v(t) = (x′(t), y′(t)). It is a vector — not a position. It tells you the direction and rate of change at each moment t

What is a critical point of a parametric curve

 A: A value of t where dy/dx = 0 (horizontal tangent) or dy/dx is undefined (vertical tangent). Found by setting the numerator or denominator of y′(t)/x′(t) equal to zero separately.

What is the formula for dy/dx of a parametric curve?

A: dy/dx = y′(t) / x′(t)  Derive both x and y separately with respect to t, then divide. This is the slope of the tangent line at parameter value t.

What is the formula for the second derivative d²y/dx² of a parametric curve? |

A: d²y/dx² = [d/dt (dy/dx)] / x′(t)  Differentiate dy/dx with respect to t first, then divide by x′(t) again. Do NOT compute y″(t)/x″(t) — that is wrong.

 What is the tangent line equation at parameter t₀?

| A: y − y₀ = m(x − x₀) where m = y′(t₀)/x′(t₀)  and  (x₀, y₀) = (x(t₀), y(t₀))  Compute slope, compute point, plug into point-slope form.

Q: What is the area under a parametric curve?

 A: A = ∫[a to b] y(t) · x′(t) dt  Comes from A = ∫y dx by substituting dx = x′(t) dt. Watch the direction of t — if the curve goes right-to-left, the sign flips.

Q: What is the arc length of a parametric curve from t = a to t = b?

A: L = ∫[a to b] √[(x′(t))² + (y′(t))²] dt  This is just speed integrated over time — total distance traveled along the curve.

Q: What is the surface area of revolution (around the x-axis) of a parametric curve?

 A: S = 2π ∫[a to b] y(t) · √[(x′(t))² + (y′(t))²] dt  For rotation around the y-axis, replace y(t) with x(t) in front of the square root.

Q: What is the key trig identity used to recognize circles and simplify speed

 A: sin²t + cos²t = 1  Always. It comes from the Pythagorean theorem on the unit circle. Use it to eliminate t from x²+y² and to simplify √[A²sin²t + A²cos²t] = A.

Q: What is the power-reducing identity for cos²t? When do you use it?

 A: cos²t = (1 + cos 2t) / 2  Use it whenever you need to integrate cos²t — there is no direct antiderivative for cos², so you must rewrite first. Same pattern: sin²t = (1 − cos 2t) / 2.

A velocity vector is written as (4t, 1). Is that a position or a direction?

A: Direction — not a position. The notation (a, b) for a vector means "move a units horizontally and b units vertically." It is an instruction for motion, not a location. Same notation as a coordinate point, completely different meaning.

What is the difference between the (x, y) table values and the (x′, y′) table values when plotting?

A: (x, y) values are positions — the actual dots you connect to draw the curve. (x′, y′) values are velocity vectors — arrows showing direction of motion at each position. You never plot x′,y′ as positions; you draw them as arrows from each (x,y) point.

Q: How do you read the direction of motion from a velocity vector without plotting?

A: Read the sign of each component: • x-component positive → moving right; negative → moving left • y-component positive → moving up; negative → moving down The magnitude tells you how fast in that direction. No plotting needed.

What are the two cases for critical points and what do they mean geometrically?

A: Case A: set y′(t) = 0 → numerator is zero → dy/dx = 0 → horizontal tangent line. Case B: set x′(t) = 0 → denominator is zero → dy/dx undefined → vertical tangent line. These are the only two ways a fraction can be zero or undefined.

 Why does a circle of radius R have constant speed R in parametric form?

A: For x = R cos t, y = R sin t: Speed = √[(−R sin t)² + (R cos t)²] = √[R²sin²t + R²cos²t] = √[R²(sin²t + cos²t)] = √[R²·1] = R The Pythagorean identity collapses everything to R for all t.

How do you recognize that (A cos t, A sin t) traces a circle?

 A: Same nonzero coefficient A on both, one has cosine and one has sine. When you compute x² + y² = (A cos t)² + (A sin t)² = A²(cos²t + sin²t) = A², you get x² + y² = A² — a circle of radius A.

What does it mean if dy/dx = 0 but x′(t) = 0 at the same t value?

A: Both numerator and denominator are zero — this is the indeterminate form 0/0. You cannot conclude horizontal or vertical tangent without further analysis. This is a special case requiring more work (L'Hôpital or limit analysis).

Why does arc length equal ∫speed dt?

A: Each tiny arc piece has length √(dx² + dy²) by the Pythagorean theorem. Dividing by dt gives √[(dx/dt)² + (dy/dt)²] which is exactly the speed formula. Integrating speed over time gives total distance — same intuition as distance = rate × time.

 Common mistake: computing d²y/dx² as y″(t) / x″(t). Why is this wrong?

A: Because d²y/dx² means differentiating dy/dx with respect to x — not differentiating y and x separately twice. The correct formula is: d²y/dx² = [d/dt(dy/dx)] / x′(t) You differentiate the first derivative (which is a function of t) with respect to t, then divide by x′(t) once more.

Common mistake: forgetting the minus sign when differentiating cos t. What is the correct derivative?

 A: d/dt [cos t] = −sin t  The minus sign is mandatory. Forgetting it flips every sign in your velocity vector and slope calculation. Say it out loud: "derivative of cosine is negative sine."