Questions from General Calculus

Q: Illustrate l’Hospital’s Rule by graphing both f (x)/g(

Illustrate l’Hospital’s Rule by graphing both f (x)/g(x) and f ’(x)/g’ (x) near x = 0 to see that these ratios have the same limit as x ( 0. Also, calculate the exact value of the limit. f (x) = 2x si...

Q: (a) Where does the normal line to the ellipse x2

(a) Where does the normal line to the ellipse x2 - xy + y2 = 3 at the point (-1, 1) intersect the ellipse a second time? (b) Illustrate part (a) by graphing the ellipse and the normal line.

Q: Suppose f (3) = 2, f ‘(3)

Suppose f (3) = 2, f ‘(3) = 1/2 , and f ’(x) > 0 and f ’’(x) < 0 for all x. (a) Sketch a possible graph for f. (b) How many solutions does the equation f (x) = 0 have? Why? (c) Is it possible that f ‘...

Q: Investigate the family of curves f (x) = ex -

Investigate the family of curves f (x) = ex - cx. In particular, find the limits as x ( ±∞ and determine the values of c for which f has an absolute minimum. What happens to the minimum points as c in...

Q: Suppose f is a continuous function where f (x) >

Suppose f is a continuous function where f (x) > 0 for all x, f (0) = 4, f ‘(x) > 0 if x < 0 or x > 2, f ‘(x) < 0 if 0 < x < 2, f ’’(-1) = f ’’(1) = 0, f ’’(x) > 0 if x < -1 or x > 1, f ’’(x) < 0 if -...

Q: Find all points on the curve x2y2 + xy = 2 where

Find all points on the curve x2y2 + xy = 2 where the slope of the tangent line is -1.

Q: Let / (a) Use the definition of derivative

Let (a) Use the definition of derivative to compute f ‘(0). (b) Show that f has derivatives of all orders that are defined on R.

Q: Let / (a) Show that f is continuous

Let (a) Show that f is continuous at 0. (b) Investigate graphically whether f is differentiable at 0 by zooming in several times toward the point (0, 1) on the graph of f . (c) Show that f is not diff...

Q: The graph of a function y = f (x) is

The graph of a function y = f (x) is shown. At which point(s) are the following true? (a) dy/dx and d2y/dx2 are both positive. (b) dy/dx and d2y/dx2 are both negative. (c) dy/dx is negative but d2y/dx...

Q: Find equations of both the tangent lines to the ellipse x2 +

Find equations of both the tangent lines to the ellipse x2 + 4y2 = 36 that pass through the point (12, 3).