Chapter 14

The position function of a particle moving on a coordinate line is given as \(s(t)=t^{2}-6 t-7,0 \leq t \leq 10 .\) Find the displacement and total distance traveled by the particle from \(1 \leq t \leq 4\).

The velocity function of a moving particle on a coordinate line is \(v(t)\) \(=2 t+1\) for \(0 \leq t \leq 8 .\) At \(t=1,\) its position is \(-4 .\) Find the position of the particle at \(t=5\).

The rate of depreciation for a new piece of equipment at a factory is given as \(p(t)=50 t-600\) for \(0 \leq t \leq 10,\) where \(t\) is measured in years. Find the total loss of value of the equipment over the first 5 years.

If the acceleration of a moving particle on a coordinate line is \(a(t)=\) -2 for \(0 \leq t \leq 4,\) and the initial velocity \(v_{0}=10,\) find the total distance traveled by the particle during \(0 \leq t \leq 4\).

If oil is leaking from a tanker at the rate of \(f(t)=10 e^{0.2 t}\) gallons per hour where \(t\) is measured in hours, how many gallons of oil will have leaked from the tanker after the first 3 hours?

The change of temperature of a cup of coffee measured in degrees Fahrenheit in a certain room is represented by the function \(f(t)=-\cos \left(\frac{t}{4}\right)\) for \(0 \leq t \leq 5,\) where \(t\) is measured in minutes. If the temperature of the coffee is initially \(92^{\circ} \mathrm{F},\) find its temperature after the first 5 minutes.

If the half-life of a radioactive element is 4500 years, and initially, there are 100 grams of this element, approximately how many grams are left after 5000 years?

Find a solution of the differential equation: \(\frac{d y}{d x}=x \cos \left(x^{2}\right) ; y(0)=\pi\)

If \(\frac{d^{2} y}{d x^{2}}=x-5\) and at \(x=0, y^{\prime}=-2\) and \(y=1,\) find a solution of the differential equation.

Find the average value of \(y=\tan x\) from \(x=\frac{\pi}{4}\) to \(x=\frac{\pi}{3}\).

The sales of an item in a company follow an exponential growth/decay model, where \(t\) is measured in months. If the sales drop . from 5000 units in the first month to 4000 units in the third month, how many units should the company expect to sell during the seventh month?

Find an equation of the curve that has a slope of \(\frac{2 y}{x+1}\) at the point ( \(x\), \(y\) ) and passes through the point (0,4) .

The population in a city was approximately 750,000 in 1980 , and grew at a rate of \(3 \%\) per year. If the population growth followed an exponential growth model, find the city's population in the year 2002 .

Find a solution of the differential equation \(4 e^{y}=y^{\prime}-3 x e^{y}\) and \(y(0)=\) \(0 .\)

How much money should a person invest at \(6.25 \%\) interest compounded continuously so that the person will have \(\$ 50,000\) after 10 years?

The velocity function of a moving particle is given as \(v(t)=2-6 e^{-t}\), \(t \geq 0\) and \(t\) is measured in seconds. Find the total distance traveled by the particle during the first 10 seconds.

Evaluate \(\int_{0}^{1} \frac{x^{2}}{x^{3}+1} d x\).

(Calculator) The slope of a function \(y=f(x)\) at any point \((x, y)\) is \(\frac{y}{2 x+1}\) and \(f(0)=2\). (a) Write an equation of the line tangent to the graph of \(f\) at \(x=0\). (b) Use the tangent in part (a) to find the approximate value of \(f\) (0.1) (c) Find a solution \(y=f(x)\) for the differential equation. (d) Using the result in part (c), find \(f(0.1)\).