Determine whether S is a basis for $P_3$. $S=\left\{1-2 t^{2}+t^{3},-4+t^{2}, 2 t+t^{3}, 5 t\right\}$

Use the definite integral to find the area between the x-axis and f(x) over the indicated interval. Check first to see if the graph crosses the x-axis in the given interval. $f(x)=x^{3}$; [-1, 3]

Decide whether or not the given set constitutes a vector space. Assume "standard " definitions of the operations. The set of all continuous functions $f$ defined on the interval [0, 1] such that $f(0)=1$

Find an arithmetic progression containing ten primes that begins with 199.

Johanna is planting tomatoes in the school garden this year. Tomato plants come in packs of six. She needs 80 plants in the garden and already has 28. How many packs of plants will she need?

(a) Write a structural formula for a sulfide having the molecular formula $\mathrm{C}_{3} \mathrm{H}_{8} \mathrm{S}$. (b) What two thiols have the molecular formula $\mathrm{C}_{3} \mathrm{H}_{8} \mathrm{S}$?

In Springfield the population P grows at the rate of 2% per year. The equation $P=1,500,000(1.02)^{t}$ gives the population t years after 2015. Find the value of t for which the tenth of a year.

Find each integral. Each integral contains a term of the form $\sqrt{a^{2}-x^{2}}$. $\int \sqrt{4-x^{2}} d x$.

Determine whether the system has infinitely many solutions or only the trivial solution. Do not solve the systems.

$$\left\{\begin{array}{l}{3 x+2 y+z=0} \\ {2 x+2 y+z=0} \\ {4 x+y+z=0}\end{array}\right.$$

At what wavelength does a cavity at $6000^{\circ} \mathrm{K}$ radiate most per unit wavelength?

For what values of does the function $y=e^{r x}$ satisfy the equation

You read on a financial website that the nominal interest rate is 12 percent per year in Canada and 8 percent per year in the United States. Suppose that international capital flows equalize the real interest rates in the two countries and that purchasing-power parity holds.

Using the Fisher equation, what can you infer about expected inflation in Canada and in the United States?

What can you infer about the expected change in the exchange rate between the Canadian dollar and the U.S. dollar?

Find each definite integral. $\int_{3}^{0}(-\pi) d t$.

Rationalize the denominators. $\frac{6}{\sqrt{5}}$