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Jacob Smith

Jacob S.

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Construct a function that passes through the origin with a constant slope of 1, with removable discontinuities at $x = ? 7$ and $x = 1$.

Construct a function that passes through the origin with a constant slope of 1, with removable discontinuities at $x = ? 7$ and $x = 1$.

Precalculus

Introduction to Calculus

Continuity
For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and righthand limits of the function given as $x$ approaches $a$. If the function has a limit as $x$ approaches $a$, state it. If not, discuss why there is no limit. $$\lim _{x \rightarrow 0} \frac{5}{1-e^{\frac{2}{x}}}$$

For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and righthand limits of the function given as $x$ approaches $a$. If the function has a limit as $x$ approaches $a$, state it. If not, discuss why there is no limit. $$\lim _{x \rightarrow 0} \frac{5}{1-e^{\frac{2}{x}}}$$

Precalculus

Introduction to Calculus

Finding Limits: Numerical and Graphical…

Questions asked

ANSWERED

Mary Wakumoto verified

Numerade educator

The spacecraft is nearly ready for takeoff! The last task is to attach the start capacitor to the ship in a simple RC-circuit. Your computer warns you that some adjustments may need to be made if the heat generated in the resistor over the initial time interval is higher than a given threshold. Your computer is able to provide the following information, along with a note that the units are already accounted for in the equations: Heat Generated in Resistor over the first 2 seconds: ( Q=frac{V_{b}^{2}}{R} int_{0}^{2} e^{-2 t /(R C)} d t ) joules Voltage of the Battery: ( V_{b}=5 ) volts Resistance: ( R=5 ) ohms Capacitance: ( C=frac{1}{5} ) farads Maximum Heat Allowed in Resistor: ( Q_{max }=2.5 ) joules Is the heat generated in the resistor over the first 2 seconds below the maximum allowable threshold? Yes No What is the difference between the maximum threshold (provided above by your computer) and the amount of heat generated in the resistor (that you calculated)? Provide a positive value up to three decimal places. Number joules

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ANSWERED

Adeline Hilliard verified

Numerade educator

With access to the spacecraft's controls, you connect your computer and begin installing the fuel tank and start capacitor. Your computer informs you that it can plot most of the course to your home planet but needs assistance with determining your velocity and vertical position at a critical moment in time. Your computer provides you with a function that models your acceleration during the first few seconds of flight. You will need to take this information and calculate your velocity and vertical position at time t = 2 s. Your computer notes that you can use the below initial conditions in supporting your calculations: Initial time: 0 s Acceleration during this time: a(t) = 68t³ + 59t m/s² What values (to 2 decimal places) do you provide your computer? v(2) = Number m/s s(2) = Number m

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ANSWERED

Vincenzo Zaccaro verified

Numerade educator

[ int_{-5}^{8} f(x) mathrm{d} x=9 ] [ int_{-5}^{8} g(x) mathrm{d} x=6 ] [ int_{-5}^{6} g(x) mathrm{d} x=3 ] First Number: ( int_{-5}^{8}(-4 f(x)+2 g(x)) mathrm{d} x ) Second Number: ( int_{8}^{-5}(-4 f(x)+2 g(x)) mathrm{d} x+int_{8}^{8} f(x) mathrm{d} x= ) Third Number: ( int_{6}^{8} 2 g(x) mathrm{d} x= )

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ANSWERED

Vincenzo Zaccaro verified

Numerade educator

Having stabilized the fuel, you look to your computer for next steps. You are told that the spacecraft uses a "start capacitor" that must reach a required voltage in order to start its engine. This voltage is determined by the fixed capacitance and charge as a function of time, ( q(t) ). These capacitors are charged and stored in your current location and must be brought to the spacecraft. Your computer provides you with the following definitions: Charge: The charge as a function of time is ( q(t)=int i(t) d t ), where ( i(t)=30 t^{5}+6 t^{2}-6 ) is the current as a function of time. You need to provide the computer with the function ( q(t) ) in order for it to appropriately charge the start capacitor. If an unknown constant is needed, it can be included using "+ ( mathrm{C} ) ", being sure to use a capital letter as the lowercase is used for capacitance. [ q(t)= ]

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INSTANT ANSWER

Having finalized the container, you are notified by your computer that you need to mix a stabilizer with the fuel to ensure that it survives the time to the spacecraft. The computer provides a readout of the current stabilization level along with minimum and maximum levels of stabilization. An equation, \( s(t) \), that models the levels of stabilization over time is also provided. You only need to consider these stabilization levels as \( t \) gets very large (approaches \( \infty \) ). \[ s(t)=\frac{\ln (3 / t+1)}{2 / t} \] Minimum Level of Stabilization: 1.1925 Maximum Level of Stabilization: 2.73 If the stabilization level will exceed the maximum allowed level, you must add a destabilizer. If the stabilization level will fall below the minimum allowed level, you need to add more stabilizer. Finally, if the stabilization level is between these bounds, no action is required. What do you do? Add Stabilizer Do Nothing Add Destabilizer

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ANSWERED

Bcrypt_Sha256$$2B$12$Koudzt7Vugfesdqzt.Btsohdsno3/5Wc5Bsgjhyqjgxswzij15Z06 Bcrypt_Sha256$$2B$12$Koudzt7Vugfesdqzt.Btsoec1F7Nikxndin/Owbntbjji9Jzcznki verified

Numerade educator

You see a warning notice that supplies of materials for constructing this container are low. In order to create the container with limited supplies, you will need to provide the height and radius (to 2 decimal places) that minimizes the surface area of the container. What values do you provide? Total Volume of Fuel: ( 19 pi ) Equation for Volume of this Container: ( V=pi r^{2}left(h+frac{2}{3} r ight) ) Equation for the Surface Area of this Container: ( S=2 pi r h+3 pi r^{2} ) Radius: Number Height: Number

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ANSWERED

Aishwarya Krishnakumar verified

Numerade educator

You now have the bacteria needed to create the fuel. "Computer, what is the next step in this process?" Your computer responds that the bacteria have given off enough gas (collected in a 13 L closed container) to create the fuel, and that the fuel is developed by increasing the temperature of the gas at a rate such that the pressure will initially rise at a rate of 24.9435 kPa/min (kilopascals per minute). You control the initial rate of change of temperature, and must set this value so that the initial rate of change of pressure is 24.9435 kPa/min. Once you have made this initial adjustment, the system will adjust dynamically until the fuel is created. Your computer informs you that this gas follows the Ideal Gas Law (PV = nRT), and that the units for these calculations are already accounted for. This means that no conversions are necessary in your calculations. Some initial values (at time t = 0) are provided below: Pressure (P) = 100 kPa Volume (V) = 13 L nR (a constant) = 8.3145 kPaL/K Temperature (T) = 304 K What value do you provide to your computer for the initial rate of change of temperature in kelvin/minute?

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ANSWERED

Bcrypt_Sha256$$2B$12$We1Wwocamog01O5I.V2Tkouxdh4Ofnmgpwkor7Leaonfpu0Ubfpua Bcrypt_Sha256$$2B$12$We1Wwocamog01O5I.V2Tkokttmmj7Lscvwvlptp4Rlhbswcdg9.Wy verified

Numerade educator

As the gate opens, you step into a room that appears to be a laboratory of some kind. Your computer informs you that this is where the raw materials that power the spacecraft are grown. "What do you mean, grown?", you ask. Your computer explains that a special bacteria is grown in this laboratory and converted into fuel. A specific amount of bacteria must be present for the next step in the process to work (within a reasonable margin of error). The computer provides you with information about the population of the bacteria, which follow an exponential model, shown below. In this model, ( P ) is the population of bacteria (total count, measured in millions) and ( t ) is the time measured in minutes. The population will grow based on this model once a light source is provided and stop immediately once the light source is removed. [ P(t)=7 e^{t / 2} ] Your computer continues with a note from the files that the alien civilization performed these calculations on the linearization of ( P(t) ). Therefore, you will need to linearize ( P(t) ) and then use that model to determine when to remove the light source to have 7.196 million bacteria. Your linearization should be around ( t=0 ), since this is the time when the light source is switched on. [ L(t)= ] Using this linearization, after how many minutes (to three decimal places) will you turn off the light to generate a population of 7.196 million bacteria? [ t= ext { Number minutes } ]

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ANSWERED

Kathleen Carty verified

Numerade educator

Find the absolute maximum and absolute minimum of the function ( f(x)=5 x^{2}-20 x+2 ) on the interval ( [-3,5] ). These values will be used to generate the passcode.

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ANSWERED

Kathleen Carty verified

Numerade educator

Find the open intervals on which the function ( f(x)=x+10 sqrt{2-x} ) is increasing or decreasing. The safe points will be calculated from these intervals. If the function is never increasing or decreasing, provide an input of NA to your computer.

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