Question

Use a graph to estimate the coordinates of the lowest point and the leftmost point on the curve x = t^ 4 - 7t, y = t + 2t^ 4. Then find the coordinates correct to two decimal places.

(x,y) Lowest Point = ?
(x,y) Leftmost Point=?

Answer 1

Explanation:

To estimate the coordinates of the lowest point and the leftmost point on the curve, we can plot the graph of the given equations and observe the points.

First, let's plot the graph:

To find the lowest point, we need to find the minimum value of y on the curve.

Next, let's find the derivative of y with respect to t:

dy/dt = 1 + 8t^3

To find the critical points, we set dy/dt = 0 and solve for t:

1 + 8t^3 = 0

8t^3 = -1

t^3 = -1/8

t = -0.5

Substituting t = -0.5 back into the original equations, we can find the coordinates of the lowest point:

x = (-0.5)^4 - 7(-0.5) = 0.0625 + 3.5 = 3.5625

y = (-0.5) + 2(-0.5)^4 = -0.5 + 0.0625 = -0.4375

Therefore, the coordinates of the lowest point are approximately (3.56, -0.44).

To find the leftmost point, we need to find the minimum value of x on the curve.

To do this, we can find the derivative of x with respect to t:

dx/dt = 4t^3 - 7

To find the critical points, we set dx/dt = 0 and solve for t:

4t^3 - 7 = 0

4t^3 = 7

t^3 = 7/4

t = (7/4)^(1/3) ≈ 1.34

Substituting t = 1.34 back into the original equations, we can find the coordinates of the leftmost point:

x = (1.34)^4 - 7(1.34) ≈ -6.48

y = 1.34 + 2(1.34)^4 ≈ 4.45

Therefore, the coordinates of the leftmost point are approximately (-6.48, 4.45).