Questions from General Calculus

Q: Suppose that f and t are continuous on [a, bg

Suppose that f and t are continuous on [a, bg] and differentiable on [a, b]. Suppose also that f (a) = g (a) and f’ (x) = g’ (x) for a < x < b. Prove that f (b) < g (b).

Q: Show that sin x < x if 0 < x < 2π

Show that sin x < x if 0 < x < 2π.

Q: Suppose f is an odd function and is differentiable everywhere. Prove

Suppose f is an odd function and is differentiable everywhere. Prove that for every positive number b, there exists a number c in s(-b, b) such that f ‘(c) = f (b)/b.

Q: Use the Mean Value Theorem to prove the inequality |sin

Use the Mean Value Theorem to prove the inequality |sin a - sin b | ≤ |a - b | for all a and b

Q: If f ‘(x) = c (c a constant)

If f ‘(x) = c (c a constant) for all x, use Corollary 7 to show that f (x) = cx + d for some constant d. Corollary 7:

Q: Let f (x) = 1/x and /

Let f (x) = 1/x and Show that f 9sxd − t9sxd for all x in their domains. Can we conclude from Corollary 7 that f - g is constant? Corollary 7:

Q: Use the method of Example 6 to prove the identity 2

Use the method of Example 6 to prove the identity 2 sin-1x = cos-1(1 - 2x2) x ≥ 0 Example 6: The function f (x) = |x | has its (local and absolute) minimum value at 0, but that value...

Q: At 2:00 pm a car’s speedometer reads 30 mi/

At 2:00 pm a car’s speedometer reads 30 mi/h. At 2:10 pm it reads 50 mi/h. Show that at some time between 2:00 and 2:10 the acceleration is exactly 120 mi/h2.

Q: Two runners start a race at the same time and finish in

Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same speed.

Q: Find the derivative of the function. Simplify where possible.

Find the derivative of the function. Simplify where possible. y = cos-1(sin-1 t)