An ice cube tray of negligible mass contains 0.315 kg of water at 17.7^\circ. How much heat must be removed to cool the water to 0.0^\circ and freeze it?

Calculate the integral.
${\displaystyle {\int }_0^4(6t-3)dt}$

Solve for f:
$\underset{y\to x}{lim}\frac{(x-y{)}^2}{f(x)-f(y)}=1$

Solve $d\frac{d^{2}y}{d{t}^2}-8d\frac{dy}{dt}+15y=9t{e}^{3t}withy(0)=5,y^{\prime }(0)=10$

In there a relationship between confidence intervals and two-tailed hypothesis tests? The answer is yes. Let c be the level of confidence used to construct a confidence interval from sample data. Let * be the level of significance for a two-tailed hypothesis test. The following statement applies to hypothesis tests of the mean:
For a two-tailed hypothesis test with level of significance a and null hypothesis $H_0:\mu =k$ we reject Ho whenever k falls outside the $c=1—\alpha$ confidence interval for $\mu$ based on the sample data. When A falls within the $c=1—\alpha$ confidence interval. we do reject $H_0$.
For a one-tailed hypothesis test with level of significance Ho : $\mu =k$ and null hypothesiswe reject Ho whenever A falls outsidethe $c=1—2\alpha$ confidence interval for p based on the sample data. When A falls within the $c=1—2\alpha$ confidence interval, we do not reject $H_0$.
A corresponding relationship between confidence intervals and two-tailed hypothesis tests is also valid for other parameters, such as p, $\mu 1—{\mu }_2,and{p}_1,-p_2$.
(a) Consider the hypotheses $H_0:{\mu }_1—{\mu }_2=Oand{H}_1:{\mu }_1—{\mu }_2\ne$ Suppose a 95% confidence interval for ${\mu }_1—{\mu }_2$ contains only positive numbers. Should you reject the null hypothesis when $\alpha =0.05$? Why or why not?

A sample of 25 circuit boards yields 2 defective boards after testing. What would be confidence interval for the true proportion of circuit boards in the lot from which the sample was drawn? Use the Clopper-Pearson Interval and also a normal approximation interval. In yur own words which do feel is better to use and why?

Find the derivative of the function using shortcut rules.
${\displaystyle y=4x^2+13x+13}$