Questions from Calculus

Q: A swimmer crosses a pool of width b by swimming in a

A swimmer crosses a pool of width b by swimming in a straight line from (0, 0) to (2b, b). (See figure.) a. Let f be a function defined as the y-coordinate of the point on the long side of the pool...

Q: Find all values of c such that f is continuous on (−∞,

Find all values of c such that f is continuous on (−∞, ∞).

Q: Prove that for any real number y there exists x in (−

Prove that for any real number y there exists x in (−π/2, π/2) such that tan x = y.

Q: Let / What is the domain of f? How

Let What is the domain of f? How can you define f at x = 0 in order for f to be continuous there?

Q: Prove that if / then f is continuous at c

Prove that if then f is continuous at c.

Q: Discuss the continuity of the function h(x) = x

Discuss the continuity of the function h(x) = x[[x]].

Q: a. Let f1(x) and f2(x)

a. Let f1(x) and f2(x) be continuous on the closed interval [a, b]. If f1(a) < f2(a) and f1(b) > f2(b), prove that there exists c between a and b such that f1(c) = f2(c). b. Show that there exists c i...

Q: Sketch the graph of f′. Explain how you found your answer

Sketch the graph of f′. Explain how you found your answer.

Q: Prove or disprove: If x and y are real numbers with

Prove or disprove: If x and y are real numbers with y ≥ 0 and y(y + 1) ≤ (x + 1)2, then y(y − 1) ≤ x2.

Q: Determine all polynomials P(x) such that P(

Determine all polynomials P(x) such that P(x2 + 1) = (P(x))2 + 1 and P(0) = 0.