Questions from General Calculus

Q: Let f(x) = a1 sin x + a2 sin

Let f(x) = a1 sin x + a2 sin 2x + ∙ ∙ ∙ + an sin nx, where a1, a2, . . . , an are real numbers and n is a positive integer. If it is given that | f (x) | ≤ |sin x | for all x, show that |a1 + 2a2 + ∙...

Q: The speeds of sound c1 in an upper layer and c2 in

The speeds of sound c1 in an upper layer and c2 in a lower layer of rock and the thickness h of the upper layer can be determined by seismic exploration if the speed of sound in the lower layer is gre...

Q: If a rock is thrown vertically upward from the surface of Mars

If a rock is thrown vertically upward from the surface of Mars with velocity 15 m/s, its height after t seconds is h = 15t - 1.86t2. (a) What is the velocity of the rock after 2 s? (b) What is the vel...

Q: For what values of c is there a straight line that intersects

For what values of c is there a straight line that intersects the curve y = x4 + cx3 + 12x2 - 5x + 2 in four distinct points?

Q: One of the problems posed by the Marquis de l’Hospital in his

One of the problems posed by the Marquis de l’Hospital in his calculus textbook Analyse des Infiniment Petits concerns a pulley that is attached to the ceiling of a room at a point C...

Q: Given a sphere with radius r, find the height of a

Given a sphere with radius r, find the height of a pyramid of minimum volume whose base is a square and whose base and triangular faces are all tangent to the sphere. What if the base of the pyramid i...

Q: Assume that a snowball melts so that its volume decreases at a

Assume that a snowball melts so that its volume decreases at a rate proportional to its surface area. If it takes three hours for the snowball to decrease to half its original volume, how much longer...

Q: Show that |sin x - cos x | ≤ 2

Show that |sin x - cos x | ≤ 2 for all x.

Q: Does the function f (x) = e10| x-

Does the function f (x) = e10| x-2| - x2 have an absolute maximum? If so, find it. What about an absolute minimum?

Q: Show that x2y2 (4 - x2)(4 - y2)

Show that x2y2 (4 - x2)(4 - y2) ≤ 16 for all numbers x and y such that |x | ≤ 2 and |y | ≤ 2.