Natural exponential Function

In summary: You should find the derivative of sin(e^x), set it equal to 0 to find the critical points, and make sure those occur between 3.8 and 4. Then check the values of sin(e^x) at the critical points and at the endpoints.
  • #1
cnc86666
1
0
1. Let f(x)=sin(e^x)

a. Find 2 values of x satisfying f(x)=0

b. What is the range of f(x)

c. Find the value(s) of x that maximize f on [3.8,4] (use calculus)



2. y=e^x if and only if x=ln y



3.a. x=-infinity because the limit of e^x as x approaches -infinity is 0. and also, x=infinity because limit e^x as a approaches infinity=infinity

b.The range of f(x) is (0,infinity)

c. the values of x that maximize f on 3.81 and 3.95
 
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  • #2
It might be better in this format:

Homework Statement


Let f(x)=sin(e^x)

a. Find 2 values of x satisfying f(x)=0

b. What is the range of f(x)

c. Find the value(s) of x that maximize f on [3.8,4] (use calculus)

Homework Equations


y=e^x if and only if x=ln y

The Attempt at a Solution


a. x=-infinity because the limit of e^x as x approaches -infinity is 0. and also, x=infinity because limit e^x as a approaches infinity=infinity

b.The range of f(x) is (0,infinity)

c. the values of x that maximize f on 3.81 and 3.95[/QUOTE]
 
  • #3
Thanks, micromass.

(a) "infiinity" and "-infinity" are NOT "values of x" so I doubt those are acceptable answers. sin(y)= 0 for y any multiple of [itex]\pi[/itex]. You just need to find x such that [itex]e^x= \pi[/itex] and, say, [itex]e^x= 2\pi[/itex].

(b) Why does the range not include negative numbers? [itex]sin(3\pi/2)= -1[/itex] and there certainly exist x such that [itex]e^x= 3\pi/2[/itex]. And how could [itex]sin(e^x)[/itex] be larger than 1?

(c) I get more than just two points.
 

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, approximately equal to 2.718. It is a special type of exponential function that models exponential growth or decay in a variety of natural systems.

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

The main difference between the natural exponential function and other exponential functions is the base. While the natural exponential function has a base of e, other exponential functions can have any positive base, such as 2, 10, or 100. Additionally, the natural exponential function has a special relationship with the natural logarithm, making it easier to solve for x in equations involving both functions.

3. What are some real-world applications of the natural exponential function?

The natural exponential function is used to model a variety of natural phenomena, such as population growth, radioactive decay, and compound interest. It is also used in fields such as physics, chemistry, and biology to describe processes that exhibit exponential growth or decay.

4. How can the natural exponential function be graphed?

The graph of the natural exponential function is a smooth, continuously increasing curve that passes through the point (0,1) on the coordinate plane. The curve approaches but never reaches the x-axis, showing that the function grows without bound as x increases. The rate of growth increases as x increases, resulting in a steep curve that becomes nearly vertical as x approaches infinity.

5. What is the inverse of the natural exponential function?

The inverse of the natural exponential function is the natural logarithm, denoted as ln(x). This means that if y = e^x, then x = ln(y). In other words, the natural logarithm "undoes" the exponential function, allowing us to solve for the original input (x) when given the output (y).

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