Find the points on the curve where the tangent is

Question

1. Find dy/dx and d²y/dx². For which values of t is the curve concave upward?

x=t³ + 1, y = t² - t  

2. Find the points on the curve where the tangent is horizontal or vertical.

x = t³ - 3t,  y = t³ - 3t²

3. Find the equation of the tangents to the curve

 x = 3t² + 1,  y = 2t³ + 1  , that pass through the  point (4, 3).

4. Find the area enclosed by the x-axis and the curve

x = 1 + e^t,  y = t - t².

5. Find the length of the loop of the curve

x = 3t - t³,   y = 3t².

6. Sketch the curve and find the area that it encloses.

 r = 4 + 3sinθ

7. Find the area of the region that lies the curve r = 2 + sinθ and outside the curve r = 3sinθ.

8. Find the area of the region that lies inside both curves.

r = 1 + cosθ,  r = 1 - cosθ

9. Find the exact length of the polar curve.

r = 2(1 + cosθ)

Answer 1

As the given data which signify about to tangent of horizontal or vertical 1. Discover dy/dx and d2y/dx2. For which values of t is the curve concave upward? x=t3 + 1, y = t2 - t 2. Discover the points on the curve where the tangent is horizontal or vertical. x = t3 - 3t, y = t3 - 3t2 3. Discover the equation of the tangents to the curve x = 3t2 + 1, y = 2t3 + 1 , that pass through the point (4, 3). 4. Discover the area encircled by the x-axis and the curve x = 1 + et, y = t - t2. 5. Discover the length of the loop of the curve x = 3t - t3, y = 3t2. 6. Sketch the curve and discover the area that it encloses. r = 4 + 3sin?