Use the recursion relations to obtain the following recursion relations for spherical Bessel functions. We have written them for $j_n$, but they are valid for yn and for the $h_n’s$. $(d / d x)\left[x^{n+1} j_{n}(x)\right]=x^{n+1} j_{n-1}(x)$

To find the following absolute values. $\left|2 e^{i \pi / 6}\right|^{2}$

Find the inverse of each function that is one-to-one. $f(x)=x^{3}-3$

Use the methods introduced in the sections to evaluate the following integrals. $\int_{0}^{2} \frac{d x}{\sqrt[3]{|x-1|}}$

Use a graphing calculator to find the coordinates of the turning points of the graph of each polynomial function in the given domain interval. Give answers to the nearest hundredth. $f(x)=2 x^{3}-5 x^{2}-x+1$; [1.4, 2]

What is the value of each of these prefix expressions? a) − ∗ 2 / 8 4 3 b) ↑ − ∗ 3 3 ∗ 4 2 5 c) + − ↑ 3 2 ↑ 2 3 / 6 − 4 2 d) ∗ + 3 + 3 ↑ 3 + 3 3 3

A point moves in the (x, y) plane on the line $2x+3y −4 = 0.$ Where will it be when the sum of the squares of its distances from (1, 0) and (−1, 0) is smallest?

Find the general solution of the following differential equations (complementary function + particular solution). Find the particular solution by inspection. Also find a computer solution and reconcile differences if necessary, noticing especially whether the particular solution is in simplest form. $\left(D^{2}+2 D+17\right) y=60 e^{-4 x} \sin 5 x$

Summarize all pertinent information obtained by applying the graphing strategy, and sketch the graph of y=f(x). $f(x)=\frac{x^{3}-5 x^{2}+6 x}{x^{2}-x-2}$

How was Maria Van Angola able to remain legally free, even under English rule?

Find the domain of each function. Write answers using interval notation. $f(x)=\sqrt{(x-3)(x+2)(x-4)}$

Find $f^{\prime}(x)$ and simplify. $f(x)=x^{3} \ln x$

You are working in a laboratory that studies the effects of currents in various crystals. One of the experiments involves a requirement for a steady current of I = 0.500 A in a wire that delivers the current to the crystal. Both the wire and the crystal are in a chamber whose interior temperature T will vary from $-40.0^{\circ} \mathrm{C}$ to $150^{\circ} \mathrm{C}$. The wire is made of tungsten and is of length L = 25.0 cm and radius r = 1.00 mm A test run is being made before the crystal is added to the circuit. Your supervisor asks you to determine the range of voltages that must be supplied to the wire in the test run to maintain its current at 0.500 A.

Suppose point A has coordinates (5, -3). What is the equation of the horizontal line through A?