Which of the following functions is a solution to the differential equation $y^{\prime }=2y$?

In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.

g(x) = { x²/³, x ≥ 0

x¹/³, x < 0

In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.

g(x) = { 2x − x³ − 1, x ≥ 0

x − (1 / (x + 1)), x < 0

b. Show that

f(x) = { x² sin(1/x), x ≠ 0

0, x = 0

is differentiable at x = 0 and find f′(0).

Theory and Examples

In Exercises 51 and 52, give reasons for your answers.

Let f(x) = |x³ − 9x|.

a. Does f'(0) exist?

Let $f$ be a differentiable function such that $f\left(2\right)=5$ and $f^{\prime }\left(2\right)=3$. What is the value of the derivative of $g\left(x\right)=2f\left(x\right)$ at $x=2$?

Which of the following functions is a solution to the differential equation $\frac{dy}{dx}=e^{2x}+2y$?

Which of the following statements is true about the differential form $(y^2-2xy)dx+(x^2-2xy)dy$ on the plane, and what is the value of the line integral of this form along any path from $(0,0)$ to $(1,1)$?

Let $f$ be a twice-differentiable function such that $f\left(0\right)=1$, $f^{\prime }\left(0\right)=2$, and $f^{″}\left(0\right)=3$. What is the value of the second-degree Taylor polynomial of $f$ centered at $x=0$ evaluated at $x=1$?