Solve over the Interval cos( square root of x)=e^x-2 , (0,1)
The problem is asking for the solution(s) to the equation cos(√x) = e^x - 2 within the interval (0,1). This is a transcendental equation involving a trigonometric function, cosine, and the exponential function, e^x, where e is the base of the natural logarithm. The equation is set within the real number interval from 0 to 1, excluding the endpoints. The task is to find the value or values of x that satisfy the equation within that specific range.
$cos \left(\right. \sqrt{x} \left.\right) = e^{x} - 2$,$\left(\right. 0 , 1 \left.\right)$
Answer
Solution:
Step 1:
Plot the functions $y = \cos(\sqrt{x})$ and $y = e^x - 2$ on the same graph. The x-coordinate where the graphs intersect represents the solution. Numerically, this is found to be $x \approx 0.94190814$.
Step 2:
Verify that the solution lies within the given range $(0, 1)$. The calculated solution $x = 0.94190814$ indeed falls within this interval.
Step 3:
Conclude that the solution to the equation $\cos(\sqrt{x}) = e^x - 2$ within the interval $(0, 1)$ is $x = 0.94190814$.
Knowledge Notes:
To solve the equation $\cos(\sqrt{x}) = e^x - 2$ over the interval $(0,1)$, we can use graphical or numerical methods since an analytical solution may not be straightforward. Here are the relevant knowledge points:
Graphical Representation: By graphing both sides of the equation, we can visually inspect for points of intersection. The x-coordinate of the intersection point(s) will be the solution(s) to the equation.
Cosine Function: The cosine function, denoted as $\cos(x)$, is a periodic function with a period of $2\pi$ and ranges from $-1$ to $1$. In this problem, we are dealing with $\cos(\sqrt{x})$, which means we take the square root of the variable before applying the cosine function.
Exponential Function: The function $e^x$ represents an exponential growth where the base is Euler's number (approximately 2.71828). In the equation, we subtract 2 to get $e^x - 2$.
Interval Checking: After finding a potential solution, it's important to check that it lies within the specified interval. In this case, the interval is $(0, 1)$, which means we are only interested in solutions where $0 < x < 1$.
Numerical Approximation: When an equation cannot be solved analytically, numerical methods such as Newton's method, bisection method, or using a graphing calculator can be employed to approximate the solution to a desired degree of accuracy.
LaTeX Formatting: When presenting mathematical equations or data, LaTeX is used to format the expressions for clarity and precision. In this solution, LaTeX is used to properly display the mathematical functions and the interval notation.
