Questions from General Calculus

Q: Suppose that we replace the parabolic mirror of Problem 18 by a

Suppose that we replace the parabolic mirror of Problem 18 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...

Q: Given an ellipse x2/a2 + y2/b2 = 1

Given an ellipse x2/a2 + y2/b2 = 1, where a ≠ b, find the equation of the set of all points from which there are two tangents to the curve whose slopes are (a) reciprocals and (b) negative reciprocals...

Q: Find the two points on the curve y = x4 – 2x2

Find the two points on the curve y = x4 – 2x2 – x that have a common tangent line.

Q: Suppose that three points on the parabola y = x2 have the

Suppose that three points on the parabola y = x2 have the property that their normal lines intersect at a common point. Show that the sum of their x-coordinates is 0.

Q: The equation x2 – xy + y2 = 3 represents a “

The equation x2 – xy + y2 = 3 represents a “rotated ellipse,” that is, an ellipse whose axes are not parallel to the coordinate axes. Find the points at which this ellipse crosses the -axis and show t...

Q: A lattice point in the plane is a point with integer coordinates

A lattice point in the plane is a point with integer coordinates. Suppose that circles with radius r are drawn using all lattice points as centers. Find the smallest value of r such that any line with...

Q: Suppose f is a function with the property that |f (

Suppose f is a function with the property that |f (x)

Q: The graph of f is given. /

The graph of f is given. (a). Find each limit, or explain why it does not exist. (b). State the equations of the horizontal asymptotes. (c). State the equations of the vertical asymptotes. (d). At w...

Q: Use graphs to discover the asymptotes of the curve. Then prove

Use graphs to discover the asymptotes of the curve. Then prove what you have discovered.

Q: Let / (a). Evaluate each limit,

Let (a). Evaluate each limit, if it exists. (b). Where is f discontinuous? (c). Sketch the graph of f.