Skip to main content
Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblema 12.1.67b
Capitolo 12, Problema 12.1.67b

67–72. Derivatives Consider the following parametric curves.
b. Make a sketch of the curve showing the tangent line at the point corresponding to the given value of t.


x = 2 + 4t, y = 4 − 8t; t = 2

Guida verificata passo dopo passo
1
Identify the parametric equations given: \(x = 2 + 4t\) and \(y = 4 - 8t\).
Find the coordinates of the point on the curve at \(t = 2\) by substituting \(t = 2\) into both equations: calculate \(x(2)\) and \(y(2)\).
Compute the derivatives of \(x\) and \(y\) with respect to \(t\): find \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\).
Determine the slope of the tangent line at \(t = 2\) using the formula for parametric curves: \(\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}\) evaluated at \(t = 2\).
Write the equation of the tangent line in point-slope form using the point \((x(2), y(2))\) and the slope found in the previous step: \(y - y(2) = m (x - x(2))\).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
3m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, usually t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves and motions.
Video consigliato:
08:02
Parameterizing Equations

Derivative of Parametric Curves

To find the slope of the tangent line to a parametric curve, compute dx/dt and dy/dt, then find dy/dx = (dy/dt) / (dx/dt). This derivative gives the instantaneous rate of change of y with respect to x at a given parameter value.
Video consigliato:
Percorso guidato
06:49
Differentiation of Parametric Curves

Equation of the Tangent Line

Once the slope of the tangent line at a point is known, use the point-slope form y - y₀ = m(x - x₀) to write the tangent line's equation. Here, (x₀, y₀) is the point on the curve at the given parameter t, and m is the slope.
Video consigliato:
Percorso guidato
05:14
Equations of Tangent Lines
Pratica correlata
Domanda del libro di testo

Volume of a hyperbolic cap Consider the region R bounded by the right branch of the hyperbola x²/a² - y²/b² = 1 and the vertical line through the right focus.

b. What is the volume of the solid that is generated when R is revolved about the y-axis?

62
views
Domanda del libro di testo

Tangents and normals: Let a polar curve be described by r = f(θ), and let ℓ be the line tangent to the curve at the point P(x,y) = P(r,θ) (see figure).

b. Explain why tan θ = y/x.


106
views
Domanda del libro di testo

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


c. The parametric equations x=t, y=t², for t≥0, describe the complete parabola y=x².

31
views
Domanda del libro di testo

Intersecting lines Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.


b. x = 2 + 5s, y = 1 + s and x = 4 + 10t, y = 3 + 2t

40
views
Domanda del libro di testo

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


b. An object following the parametric curve x=2cos 2πt, y=2 sin 2πt circles the origin once every 1 time unit.

35
views
Domanda del libro di testo

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. On every ellipse, there are exactly two points at which the curve has slope s, where s is any real number.

32
views