# Natural exponential function, calculus

• B
• Medtner
In summary: In this case, the problem involves finding the probability of a lightbulb failing within the first 200 hours using an exponential density function with a mean of 1000 hours. After completing the integration, the calculated probability is -e^1/5 + 1, while the book's answer is 0.81. It is important to check the signals and ensure that negative probabilities are not being calculated.
Medtner
So I'm trying out various practice problems and for some reason I can't get the same answer when it comes to problems involving natural exponentials.

Here's the problem
A type of lightbulb is labeled as having an average lifetime of 1000 hours. It's reasonable to model the probability of failure of these bulbs by an exponential density function with mean μ = 1000.

a)
Use this model to find the probability that a bulb fails within the first 200 hours.

Did the work, all fine and dandy, integration and what not and it led me to this:

-e^1/5 + 1
Book says the answer is 0.81
However when I evaluate it on my calculator its -.0221402

I'm pretty sure the answer would be 0.81 if we are talking about the bulb failing after 200 hours: P(X>200).

Also, make sure you have your signals right. You can't have negative probabilities, as you probably know.

Medtner said:
So I'm trying out various practice problems and for some reason I can't get the same answer when it comes to problems involving natural exponentials.

Here's the problem
A type of lightbulb is labeled as having an average lifetime of 1000 hours. It's reasonable to model the probability of failure of these bulbs by an exponential density function with mean μ = 1000.

a)
Use this model to find the probability that a bulb fails within the first 200 hours.

Did the work, all fine and dandy, integration and what not and it led me to this:

-e^1/5 + 1
Book says the answer is 0.81
However when I evaluate it on my calculator its -.0221402

In order to get help for a calculation problem, it is useful to show each step in the calculation.

fresh_42 and ramzerimar

## 1. What is a natural exponential function?

A natural exponential function is a mathematical function of the form f(x) = e^x, where e is the base of the natural logarithm. It is commonly used in calculus and other areas of mathematics to model growth and decay phenomena.

## 2. How is the natural exponential function different from other exponential functions?

The natural exponential function is different from other exponential functions in that its base is the irrational number e, which is approximately equal to 2.718. This makes it a special case of exponential functions, and it has many unique properties and applications.

## 3. What is calculus?

Calculus is a branch of mathematics that deals with the study of change. It includes two main parts: differential calculus, which focuses on the rate of change of functions, and integral calculus, which deals with the accumulation of quantities. Calculus is widely used in fields such as physics, engineering, and economics.

## 4. How is the natural exponential function used in calculus?

The natural exponential function is used in calculus to model various growth and decay phenomena, such as population growth, radioactive decay, and compound interest. It is also used in the calculation of derivatives and integrals, making it an essential tool in solving many problems in calculus.

## 5. What are the applications of the natural exponential function in real life?

The natural exponential function has many applications in real life, including modeling the growth of bacteria and viruses, predicting population growth, and calculating the rate of decay of radioactive materials. It is also used in finance to calculate compound interest, and in physics to describe the behavior of various physical systems.

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