Find the Taylor series for the functions defined as follows. Give the interval of convergence for each series. $f(x)=x^{2} e^{x}$

Decide whether each function defined as follows is a probability density function for the given interval. $f(x)=\sqrt{x}$; [4, 9]

Currently in the American population of people aged 65 years old and older, there are approximately 130 women for every 100 men. In a nursing home with 280 male residents over the age of 65, how many female residents over the age of 65 would be expected?

Write each expression in interval notation. Graph each interval. x<4

________ variables are defined outside all functions and are accessible to any function within their scope.

Find the domain and range of the function. Write your answer in interval notation. $g(x)=\sqrt{16-x^{4}}$

Find the derivative of each function. $y=-4 \sin 7 x$

Find F(s) if: (a) $f(t)=6 e^{-t} \cosh 2 t$ (b) $f(t)=3 t e^{-2 t} \sinh 4 t$ (c) $f(t)=8 e^{-3 t} \cosh t u(t-2)$

A hydraulic lift in a garage has two pistons: a small one of cross-sectional area $4.00\ \mathrm{cm}^{2}$ and a large one of cross-sectional area $250\ \mathrm{cm}^{2} .$ (a) If this lift is designed to raise a 3500-kg car, what minimum force must be applied to the small piston? (b) If the force is applied through compressed air, what must be the minimum air pressure applied to the small piston?

An insurance policy pays for a random loss X subject to a deductible of C, where 0<C<1. The loss amount is modeled as a continuous random variable with density function

$$f(x)=\left\{\begin{array}{ll} {2 x} & {\text { for } 0<x<1} \\ {0} & {\text { otherwise }} \end{array}\right.$$

. Given a random loss X, the probability that the insurance payment is less than 0.5 is equal to 0.64. Calculate C. Choose one of the following. (Hint: The payment is 0 unless the loss is greater than the deductible, in which case the payment is the loss minus the deductible.) a. 0.1. b. 0.3. c. 0.4. d. 0.6. e. 0.8.

The amount of a radioactive substance decreases exponentially, with a decay constant of 3% per month. a. Write a differential equation to express the rate of change. b.Find a general solution to the differential equation from part a. c. If there are 75 g at the start of the decay process, find a particular solution for the differential equation from part a. d.Find the amount left after 10 months.

The following definitions and program segments has errors. Locate as many as you can. vector numbers = { 1, 2, 3, 4 };

What is the difference between an argument and a parameter variable?

Decide whether the functions defined as follows are probability density functions on the indicated intervals. If not, tell why. $f(x)=2 x^{2}$; [-1, 1]