Questions from General Calculus

Q: Show by implicit differentiation that the tangent to the ellipse x2

Show by implicit differentiation that the tangent to the ellipse x2 / a2 + y2 / b2 = 1 at the point (x0, y0) is x0x/a2 + y0y/b2 = 1

Q: At what point on the curve y = [ln(x

At what point on the curve y = [ln(x + 4)]2 is the tangent horizontal?

Q: (a) Find an equation of the tangent to the curve

(a) Find an equation of the tangent to the curve y = ex that is parallel to the line x - 4y = 1. (b) Find an equation of the tangent to the curve y = ex that passes through the origin.

Q: Find a parabola y = ax2 + bx + c that passes

Find a parabola y = ax2 + bx + c that passes through the point (1, 4) and whose tangent lines at x = -1 and x = 5 have slopes 6 and -2, respectively.

Q: An equation of motion of the form s = Ae-ct

An equation of motion of the form s = Ae-ct cos(ωt + δ) represents damped oscillation of an object. Find the velocity and acceleration of the object.

Q: Find y’’ by implicit differentiation. x2 + 4y2 = 4

Find y’’ by implicit differentiation. x2 + 4y2 = 4

Q: Let T and N be the tangent and normal lines to the

Let T and N be the tangent and normal lines to the ellipse x2/9 + y2/4 = 1 at any point P on the ellipse in the first quadrant. Let xT and yT be the x- and y-intercepts of T and xN and yN be the inter...

Q: (a) The curve with equation y2 = x3 + 3x2

(a) The curve with equation y2 = x3 + 3x2 is called the Tschirnhausen cubic. Find an equation of the tangent line to this curve at the point (1, -2). (b) At what points does this curve have horizontal...

Q: Let P(x1, y1) be a point on the

Let P(x1, y1) be a point on the parabola y2 = 4px with focus F(p, 0). Let α be the angle between the parabola and the line segment FP, and let β be the angle between the hori...

Q: Suppose that we replace the parabolic mirror of Problem 22 by a

Suppose that we replace the parabolic mirror of Problem 22 by a spherical mirror. Although the mirror has no focus, we can show the existence of an approximate focus. In the figure, C is a semicircle...