Limit, derivative, integral, and Taylor check

Problem

Evaluate $\displaystyle \lim_{x \to 0} \frac{\sin x}{x}$ . Then, for $f(x) = x^3 - 3x^2 + 2$ , compute $f'(x)$ , $\displaystyle \int_0^2 f(x)\,dx$ , and the second-order Taylor polynomial around $x=1$ .

Reveal answer or reference solution

The limit is $1$ . Also,

f'(x) = 3x^2 - 6x \qquad \text{and} \qquad \int_0^2 f(x)\,dx = 0.

Since $f(1)=0$ , $f'(1)=-3$ , and $f''(1)=0$ , the second-order Taylor polynomial is $-3(x-1)$ .

Local history

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Appears in

Calculus Practice