Evaluate $\lim_{x \to 0} \frac{\int_0^x \sin(t^2) dt}{x^3}$.
By Leibniz rule and L'Hopital: numerator derivative is $\sin(x^2), denominator$ derivative is $3x^2$. Limit = $\lim \frac{\sin(x^2)}{3x^2} = 1/3 \approx 0.333$...
Which of the following functions is strictly increasing on $\mathbb{R}$?
f'(x) = $3x^2 + 1$ > 0 for all x $\in \mathbb{R}, making f(x) strictly increasing everywhere$.
Compute $\lim_{x \to 0} \frac{x - \sin x}{x^3}$.
Using Taylor series: sin x = x - $x^3/6$ + ... so x - sin $x = x^3/6$ + ... Thus limit = 1/6 $\approx 0.1667 \approx 0.17$.
Which of the following conditions guarantee that f(x) has a local minimum at x = c?
A local minimum occurs if the first non-zero derivative at c is of even order and strictly positive.
Find the maximum value of f(x) = x(1 - x) on the interval [0, 1].
f'(x) = 1 - 2x = 0 => x = 0.5. f(0.5) = 0.5 * 0.5 = 0.25. Endpoints give 0.
