Use the method outlined as discussed previously to solve the following Bernoulli equations. a. $y^{\prime}(t)+y=2 y^2$ b. $y^{\prime}(t)-2 y=3 y^{-1}$ c. $y^{\prime}(t)+y=\sqrt{y}$

Describe and correct the error made in finding the axis of symmetry for the graph of $y=-x^2-6 x+2$.

$$\xcancel{x=\frac{-b}{2a}=\frac{-6}{2(-1)}=3}$$

Sketch the graph:\

$$y=0.5 x$$

Nitric acid and zinc react to form zine nitrate, ammonium nitrate, and water.

4Zn(s) + 10HNO$_3$(aq) $\longrightarrow$ 4Zn(NO$_3$)$_2$(aq) + NH$_4$NO$_3$(aq) + 3H$_2$O(l)

a. How many atoms of zinc react with $1.49$ g HNO$_3$?

b. Calculate the number of grams of zinc that must react with an excess of HNO$_3$ to form $29.1$ g NH$_4$NO$_3$.

Simplify the expression. $2^{4} \cdot 2^{5}-\left(2^{2}\right)^{2}$

Evaluate the logarithm without using a calculator. $\log _{8} 512$

For the differential equation

$$\begin{gather*} y' = ry - ky^2, \,\,\,\, r > 0, \,\,\,\, k > 0 \end{gather*}$$

(a) Write the Bernoulli equation in the proper form ${\dfrac{dy}{dt} + q(t) y = r(t) y^{n}}$. (b) Solve the equation. (c) Use a computer to draw several integral curves.

Show that any function of the form y= A sinh mx + B cosh mx satisfies the differential equation

$$y’’=m^2y$$

Sketch the graph of $f(x)<x+2$.

Evaluate the following without using a calculator. cos(arccos(—1))

Estimate the minimum number of subintervals needed to approximate the integral with an error of magnitude less than $10^{-4}$ by the Trapezoidal Rule.

$$\int_{-1}^1\left(x^2+1\right) d x$$

Estimate the minimum number of subintervals needed to approximate the integral with an error of magnitude less than $10^{-4}$ by the Simpson's Rule.

$$\int_0^3 \frac{1}{\sqrt{x+1}} d x$$

Suppose that $\left\{S_{1}, S_{2}\right\}$ is a partition of a set S and $x \in S.$ Which of the following are true? (a) If we know that $x \notin S_{1}$ then x must belong to $S_2.$ (b) It’s possible that $x \notin S_{1}$ and $x \notin S_{2}$ (c) Either $x \notin S_{1} \text { or } x \notin S_{2}$ (d) Either $x \in S_{1} \text { or } x \in S_{2}$ (e) It’s possible that $x \in S_{1}$ and $x \in S_{2}.$

Solve the differential equation by the method of integrating factors.

$$2 \frac { d y } { d x } + 4 y = 1$$