2. Intro to Derivatives

Use the graph of g in the figure to do the following. <IMAGE>

b. Find the values of x in (-2,2) at which g is not differentiable.

Which of the following is the general solution to the differential equation $y^{″″″}+3y^{″″}+3y^{\prime }+y=0$?

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 $(1,1)$?

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

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 $x=1$?

Determining the unknown constant Let f(x) = {2x² if x≤1 ax-2 if x>1. Determine a value of a (if possible) for which f' is continuous at x=1.

Which of the following equations represents the orthogonal trajectories of the family of curves given by $x^2+2y^2=k^2$?

Given that $\frac{dy}{dx}=2-y$ and $y=1$ when $x=1$, what is the value of $y$ when $x=2$?

Which of the following functions satisfies the equation ${f\left(x\right)}^2+{(f\text{'}(x\left)\right)}^2=1$ for all real x?

Given the system of differential equations: $\frac{dx}{dt}=z$, $\frac{dy}{dt}=x$, $\frac{dz}{dt}=y$, which of the following is the general solution for $x\left(t\right)$?

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

Differentiability and Continuity on an Interval

Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be

a. differentiable?

Give reasons for your answers.

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

Given the system of differential equations $\frac{dx}{dt}=2x+3y$ and $\frac{dy}{dt}=6x+5y$, which of the following is the general solution for $x\left(t\right)$ and $y\left(t\right)$?

a. Graph the function

ƒ(x) = { x², -1 ≤ x < 0

{ -x², 0 ≤ x ≤ 1.

b. Is ƒ continuous at x = 0?

c. Is ƒ differentiable at x = 0?

Give reasons for your answers.