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Questions and Answers
What is the half-life of the radioactive material if it disintegrates from 5 grams to 2 grams in 100 days?
What is the half-life of the radioactive material if it disintegrates from 5 grams to 2 grams in 100 days?
- 50 days
- 40 days (correct)
- 20 days
- 100 days
How much of the radioactive material remains after 60 days, starting from 5 grams?
How much of the radioactive material remains after 60 days, starting from 5 grams?
- 2.5 grams
- 3.5 grams
- 3 grams (correct)
- 4 grams
What continuous interest rate was used if the present value of $16,000 to be received in 5 years is $10,000?
What continuous interest rate was used if the present value of $16,000 to be received in 5 years is $10,000?
- 7.5%
- 8.2% (correct)
- 15%
- 10.5%
How long will it take for an initial investment to triple with continuous compounding?
How long will it take for an initial investment to triple with continuous compounding?
What is the second derivative of the function $e^{x^2}$?
What is the second derivative of the function $e^{x^2}$?
What is the integral of $ln(5x + 3) dx$?
What is the integral of $ln(5x + 3) dx$?
What is the value of $ln(7x - 1)$?
What is the value of $ln(7x - 1)$?
How can the expression $\frac{e^{7x}}{e^{e^x}}$ be transformed into the form of $Ae^{kx}$?
How can the expression $\frac{e^{7x}}{e^{e^x}}$ be transformed into the form of $Ae^{kx}$?
Explain how to find the half-life of a radioactive material using the sample decay from 5 grams to 2 grams in 100 days.
Explain how to find the half-life of a radioactive material using the sample decay from 5 grams to 2 grams in 100 days.
Describe the process to calculate the remaining amount of radioactive material after 60 days.
Describe the process to calculate the remaining amount of radioactive material after 60 days.
How do you determine the continuous interest rate from the present value of $16,000 and $10,000 over 5 years?
How do you determine the continuous interest rate from the present value of $16,000 and $10,000 over 5 years?
What steps would you take to find how many years it takes for an investment to triple using continuous compounding?
What steps would you take to find how many years it takes for an investment to triple using continuous compounding?
What technique is applied to find the second derivative of $e^{x^2}$?
What technique is applied to find the second derivative of $e^{x^2}$?
When evaluating the integral of $ln(5x + 3)$, what method would you use?
When evaluating the integral of $ln(5x + 3)$, what method would you use?
How would you derive the simplified form of the expression $ rac{e^{7x}}{e^{e^x}}$ into $Ae^{kx}$?
How would you derive the simplified form of the expression $ rac{e^{7x}}{e^{e^x}}$ into $Ae^{kx}$?
What is the significance of evaluating $ln(7x - 1)$ and its domain restrictions?
What is the significance of evaluating $ln(7x - 1)$ and its domain restrictions?
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Study Notes
Radioactive Material Disintegration
- A sample of radioactive material disintegrates from 5 grams to 2 grams in 100 days.
- You need to determine the half-life of the material, which is the time it takes for half of the material to decay.
- You also need to find out how much will remain after 60 days.
Present Value and Interest Rate
- The present value of 16,000tobereceivedin5yearsis16,000 to be received in 5 years is 16,000tobereceivedin5yearsis10,000.
- You need to determine the interest rate compounded continuously that was used to compute this present value.
- You also need to calculate how long it takes for an initial investment to triple at this interest rate.
Derivative of Exponential Function
- You need to find the second derivative of the exponential function e^(x^2).
Evaluating Integrals
- You need to evaluate the following integral:
- ∫(ln(5x + 3) + 1/(√(x))) dx.
- You also need to evaluate the following integral:
- ∫(e^(x^2))/(x^2) dx.
Evaluating Definite Integrals
- You need to evaluate the definite integral:
- ∫(ln(x^2 + e^4))/(7x - 1) dx.
- You also need to evaluate the definite integral:
- ∫(ln(x^3 + 2)) dx.
Solving Equations and Simplifying Expressions
- You need to solve the following equation for x:
- ln(x^2) - ln(2x) = -1
- You need to simplify the expression:
- (e^(x) / (e^(7x))) * 2x - 3ln(2)
- You need to write the simplified expressão in the form of Ae^(kx), where A and k are constants.
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