# calculus

Notice that the curve given by the parametric equations x=25−t^2 y=t^3−16t is symmetric about the
x-axis. (If t gives us the point (x,y),then −t will give (x,−y)).
At which x value is the tangent to this curve horizontal? x = ?
At which t value is the tangent to this curve vertical? t = ?
The curve makes a loop which lies along the x-axis. What is the total area inside the loop? Area = ?

1. 👍 1
2. 👎 0
3. 👁 1,756
1. for the horizontal tangent, you want dy/dx = 0

dy/dt = 3t^2-16
dx/dt = -2t

dy/dx = (3t^2-16)/(-2t)
dy/dx = 0 when t=±4/√3
That is, where x = 59/3

for the vertical tangent, you want dx/dy = 0, or t=0

The loop is formed where y=0. That is, t(16t^2-1) = 0
Or, t=0 or ±4

So, the area inside the loop where -4 <= t <= 4 is

∫[-4,4] y(t) dx/dt dt
= ∫[-4,4] (3t^2-16)(-2t) dt
= 256

1. 👍 1
2. 👎 2

## Similar Questions

1. ### calculus

a particle moves along the curve xy=10. if x=2 and dy/dt=3, what is the value of dx/dt. i'm guessing this is something to do with parametric equations. if x is 2, then y=5. but how do i get dx/dt. you have to differentiate the

The graph of the equation y=x^3-x is symmetric with respect to which of the following? a) the x-axis b) the y-axis c) the origin d) none of these Answer: c 2) The graph of an odd function is symmetric with respect to which of the

3. ### Calculus

Compute, in terms of A, B, h, and k, the area enclosed by the curve defined by parametric equations. x(θ)=Acosθ+h and y(θ)=Bsinθ+k. 0≤θ≤2π.?

4. ### Math

Show that the curve with parametric equations x=tcost, y=tsint, z=t lies on the cone z^2=x^2+y^2

1. ### Calculus

A curve is defined by the parametric equations: x = t2 – t and y = t3 – 3t Find the coordinates of the point(s) on the curve for which the normal to the curve is parallel to the y-axis. You must use calculus and clearly show

2. ### Math-Precalculus-Parametric Equations - Ellipses

The graph of the equation (x-h)^2/(a^2) + (y-k)^2/b^2 = 1 is an ellipse with center (h,k), horizontal axis length 2a, and vertical axis length 2b. Find parametric equations whose graph is an ellipse with center (h,k), horizontal

3. ### calculus

Find parametric equations and symmetric equations for the line through P0 and perpendicular to both given vectors. (P0 corresponds to t = 0.) P0 = (4, 3, 0) i + j and j + k Not exactly sure what to do, how do I find what is

4. ### precalculus

To what interval I must we restrict the parameter t if the graph of the parametric equations x=t+1 y = 2t^2 +1 for t in the set I is identical to the graph of the parametric equations x = 1+ sin s y = 2- cos(2s) for s in [0,pi)?

1. ### Calc 3

Find parametric equations for the tangent line to the curve of intersection of the paraboloid z = x2 + y2 and the ellipsoid 6x2 + 5y2 + 7z2 = 39 at the point (−1, 1, 2)

2. ### maths sir steve help me reiny

A curve has parametric equations x = at² , y = 2at. Find the area bounded by the curve, the x- axis, and the ordinates at t = 1 and t = 2 step plz

3. ### Math

Find parametric equations and symmetric equations for the line of intersection of the given planes. x + y + z = 2, x + z = 0 X=1 y = Z= x=, y=

4. ### Calculus 3

If the curve of intersection of the parabolic cylinder y = x^2 and the top half of the ellipsoid x^2 + 5y^2 + 5z^2 = 25. Then find parametric equations for this curve.