Differential calculus and Taylor

Compute, using the definition, the derivative of

$$f(x)=x^3$$

at a generic point $x\in \mathbb{R}$.

Solution available to those who have unlocked the subject.

Compute, using the definition, the derivative of

$$f(x)=\frac{1}{x+1}$$

at the points $x\ne -1$.

Solution available to those who have unlocked the subject.

Compute the derivative of

$$f(x)=\cos \left(x^3\right).$$

Solution available to those who have unlocked the subject.

Compute the derivative of

$$f(x)=\frac{x^2+1}{x-1}$$

at the points $x\ne 1$.

Solution available to those who have unlocked the subject.

Write the equation of the tangent line to the graph of

$$f(x)=x{e}^x$$

at the point with abscissa $x_0=0$.

Solution available to those who have unlocked the subject.

Find the points of the graph of

$$f(x)=x{e}^{-x}$$

at which the tangent line is horizontal and write the corresponding tangents.

Solution available to those who have unlocked the subject.

Determine $a\in \mathbb{R}$ so that the function

$$f(x)=\{\begin{array}{ll}\sin x& \text{if }x\le 0\\ x^2+ax& \text{if }x>0\end{array}$$

is differentiable at $x=0$.

Solution available to those who have unlocked the subject.

Find the points at which the function

$$f(x)=|x^2-4|$$

is not differentiable and classify them.

Solution available to those who have unlocked the subject.

Verify that $f(x)=x^2$ satisfies the hypotheses of Lagrange’s theorem on the interval $[1,3]$ and find the points $c$ given by the theorem.

Solution available to those who have unlocked the subject.

Find the intervals of monotonicity and the local extrema of

$$f(x)=x^4-4x^3.$$

Solution available to those who have unlocked the subject.

Compute

$$\underset{x\to 0^{+}}{\lim }{x}^x.$$

Solution available to those who have unlocked the subject.

Compute

$$\underset{x\to 0}{\lim }\frac{e^{2x}-1-2x}{x^2}.$$

Solution available to those who have unlocked the subject.

Compute

$$\underset{x\to 0}{\lim }\frac{x-\ln (1+x)-\frac{x^2}{2}}{x^3}.$$

Solution available to those who have unlocked the subject.

Write the Maclaurin polynomial of order $2$ of

$$f(x)=\sqrt{1+2x}$$

and use the Lagrange remainder to bound the error made when approximating $f\left(\frac{1}{10}\right)$ with the polynomial.

Solution available to those who have unlocked the subject.