Introductory calculus

Find $\displaystyle\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$.

Let $f(x) = 4x^3 - 5x + 2$. Compute $f'(1)$.

The tangent line to $y = x^2 - 4x$ at $x = 3$ has equation $y = mx + c$. Find $c$.

For $f(x) = x^3 - 12x$, which statement is correct?

Which function is a counterexample to the claim "if $f'(0) = 0$ then $f$ has a local maximum or minimum at $0$"?

Compute $\displaystyle\int_0^3 (x^2 - 2x)\,dx$.

In the proof of Fermat's theorem, $f$ has a local maximum at $a$. Why is $\dfrac{f(a + h) - f(a)}{h} \ge 0$ for small negative $h$?

In Lean, with f := fun x : ℝ => (x ^ 2 - 1) / (x - 1), the lesson proves both Tendsto f (𝓝[≠] 1) (𝓝 2) and f 1 = 0. What is the right conclusion?

Write right_sum(f, a, b, n) returning the right Riemann sum $\sum_{k=1}^{n} f\!\left(a + k\,\tfrac{b - a}{n}\right)\tfrac{b - a}{n}$. Use Fraction so that the sums are exact when a, b are fractions and f is a polynomial. Then check Theorem 7: for $f(x) = x^2$ on $[0, 1]$ it should equal $\tfrac{(n + 1)(2n + 1)}{6n^2}$.