Question

state whether the following statements are true or false. justify your answers.
1) lim x to 0 (1/x^2 -1/sin^2x)is in (0/0) form.
2) f(x,y)={sin(x^2y/x^3 +y^3)}/ln(x+y/x) is a homogeneous funtion of degree 2.
3) Domain of f(x,y)=xy/(x^4 +y^4) is R^2.
4) The funtion f(x,y)=(x^3y +1, x^2 +y^2) is locally invertible at (1,2).
5) The funtion f(x,y)=x^3 +y^3 is integrable on [1,2]*[1,3].

Answer 1

Answer on Question #74644 – Math – Calculus

Question

State whether the following statements are true or false. Justify your answers.

1) $\lim x$ to 0 ($1/x^2 - 1/\sin^2 x$) is in $(0/0)$ form.

2) $f(x,y) = \{\sin(x^2 y / x^3 + y^3)\} / \ln(x + y/x)$ is a homogeneous function of degree 2.

3) Domain of $f(x,y) = xy / (x^4 + y^4)$ is $R^2$.

4) The function $f(x,y) = (x^3 y + 1, x^2 y^2)$ is locally invertible at $(1,2)$.

5) The function $f(x,y) = x^3 + y^3$ is integrable on $[1,2] \times [1,3]$.

Solution

1) $\lim_{x\to 0}\left(\frac{1}{x^2} -\frac{1}{\sin^2x}\right)$ is in $\left(\frac{0}{0}\right)$ form. It is false.

$$\begin{array}{l} \lim _ {x \rightarrow 0} \left(\frac {1}{x ^ {2}} - \frac {1}{\sin^ {2} x}\right) \text{ is in } (\infty - \infty) \text{ form but} \\ \lim _ {x \rightarrow 0} \left(\frac {1}{x ^ {2}} - \frac {1}{\sin^ {2} x}\right) = \lim _ {x \rightarrow 0} \left(\frac {\sin^ {2} x - x ^ {2}}{x ^ {2} \sin^ {2} x}\right) = \left(\frac {0}{0}\right) \\ = [\text{we can use } l' \text{Hospitale rule}] = \lim _ {x \rightarrow 0} \frac {\left(\sin^ {2} x - x ^ {2}\right) ^ {\prime}}{\left(x ^ {2} \sin^ {2} x\right) ^ {\prime}} = \\ = \lim _ {x \rightarrow 0} \frac {2 \sin (2 x) - 2 x}{2 \sin (2 x) \cdot x ^ {2} + 2 x \cdot \sin^ {2} x} = \left(\frac {0}{0}\right) = [\text{we can use } l' \text{Hospitale rule}] \\ = \lim _ {x \rightarrow 0} \frac {4 \cos (2 x) - 2}{4 \cos (2 x) \cdot x ^ {2} + 2 \sin (2 x) \cdot 2 x + 2 \sin^ {2} x + 2 x \cdot 2 \sin (2 x)} = \frac {2}{0} = \infty \\ \end{array}$$

2) $f(x,y) = \frac{\sin\left(\frac{x^2y}{x^3 + y^3}\right)}{\ln\left(x + \frac{y}{x}\right)}$ is a homogeneous function of degree 2. It is false.

$$\begin{array}{l} f (\alpha x, \alpha y) = \frac {\sin \left(\frac {(\alpha x) ^ {2} \alpha y}{(\alpha x) ^ {3} + (\alpha y) ^ {3}}\right)}{\ln \left(\alpha x + \frac {\alpha y}{\alpha x}\right)} = \frac {\sin \left(\frac {\alpha^ {3} x ^ {2} y}{(x ^ {3} + y ^ {3}) \alpha^ {3}}\right)}{\ln \left(\alpha x + \frac {y}{x}\right)} = \\ = \frac {\sin \left(\frac {x ^ {2} y}{x ^ {3} + y ^ {3}}\right)}{\ln \left(\alpha x + \frac {y}{x}\right)} \neq \alpha^ {2} f (x, y) \\ \end{array}$$

3) Domain of $f(x,y) = \frac{xy}{x^4 + y^4}$ is $R^2$. It is false. $x^4 + y^4 \neq 0$. Domain of $f(x,y)$ is $R^2 / (0;0)$.

4) The function $f(x,y) = \begin{bmatrix} x^3 y + 1 \\ x^2 + y^2 \end{bmatrix}$ is locally invertible at $(1,2)$. This is true.

Jacobian matrix of $f J_{f(x,y)} = \begin{bmatrix} \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \\ \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \end{bmatrix} = \begin{bmatrix} 3x^2 y & x^3 \\ 2x & 2y \end{bmatrix}$

$$det \left(J _ {f (x, y)}\right) = \left| \begin{array}{cc} 3 x ^ {2} y & x ^ {3} \\ 2 x & 2 y \end{array} \right| = 6 x ^ {2} y ^ {2} - 2 x ^ {4}$$ $$\left. \det \left(J _ {f (x, y)}\right) \right| _ {y = 2} ^ {x = 1} = 24 - 2 = 22 \neq 0$$

5) The function $f(x,y) = x^{3} + y^{3}$ is integrable on $[1;2] \times [1;3]$. Of course this is true because $x^{3}$ is continuous function on $\mathbb{R}$, $y^{3}$ is continuous function on $\mathbb{R}$, so $f(x,y)$ is continuous function on $R^{2}$ and that's why it is integrable on $[1;2] \times [1;3]$.

Answer:

1) false;

2) false;

3) false;

4) true;

5) true.

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