-apc.2.1.06 crosses the x-axis at one point in the interval [0,1]

In summary, "-apc.2.1.06" crosses the x-axis at one point in the interval [0,1], meaning that there is only one solution to the function in this interval. This can be determined by setting the function equal to zero and solving for x. The "-apc.2.1.06" represents the function itself and can be evaluated for different values of x. A function can cross the x-axis at more than one point, indicating multiple solutions. This information can also help determine the behavior of the function in the given interval.
  • #1

karush

Gold Member
MHB
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$\tiny{APC.2.1.06}$
The graph of $y=e^{\tan x}-2$ crosses the x-axis at one point in the interval [0,1]
What is the slope of the graph at this point.
a, 0.606 b. 2 c, 2.242 d.2.961 e.3.747

[d]
$\begin{array}{rll}
\textit{given} &e^{\tan \:x}-2 &(1)\\
\textit{rewrite} &e^{\tan x}=2 &(2) \\
\textit{e thru} &\tan x=\ln 2 &(3)\\
\textit{isolate x } &x=\arctan \ln 2+\pi n &(4)\\
\textit{[0,1] } &\arctan(\ln 2)=0.606 &(5)\\
\textit{m at x} &y_m=e^{\tan (x)}\sec ^2(x) &(6)\\
\textit{m} &y_m(0.606)=2.96 &(7)
\end{array}$

it took me 2 hours to do this
must be a better way

if its correct:unsure:
 
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  • #2
fyi, this is a calculator active problem …

6695B933-7821-4557-9A00-DAC241A97F13.jpeg
 
  • #3
karush said:
$\tiny{APC.2.1.06}$
The graph of $y=e^{\tan x}-2$ crosses the x-axis at one point in the interval [0,1]
What is the slope of the graph at this point.
a, 0.606 b. 2 c, 2.242 d.2.961 e.3.747

[d]
$\begin{array}{rll}
\textit{given} &e^{\tan \:x}-2 &(1)\\
\textit{rewrite} &e^{\tan x}=2 &(2) \\
\textit{e thru} &\tan x=\ln 2 &(3)\\
\textit{isolate x } &x=\arctan \ln 2+\pi n &(4)\\
\textit{[0,1] } &\arctan(\ln 2)=0.606 &(5)\\
\textit{m at x} &y_m=e^{\tan (x)}\sec ^2(x) &(6)\\
\textit{m} &y_m(0.606)=2.96 &(7)
\end{array}$

it took me 2 hours to do this
must be a better way

if its correct:unsure:
You mean "given $e^{\tan(x)}-2= 0$". Otherwise your second line makes no sense!
 
  • #4
ok guess I was assuming that
skeeter said:
fyi, this is a calculator active problem …

View attachment 11165
what graphing program is that from?
doesnt look like Desmos
 
  • #5
karush said:
ok guess I was assuming that

what graphing program is that from?
doesnt look like Desmos

Good Grapher Pro app on my ipad.
 

What does the expression "-apc.2.1.06 crosses the x-axis at one point in the interval [0,1]" mean?

This expression refers to a mathematical function, where the value of the function equals zero at a specific point on the x-axis within the interval of 0 to 1.

What is the significance of a function crossing the x-axis at one point?

When a function crosses the x-axis at one point, it means that there is only one solution to the equation. This can be useful in solving problems or finding the roots of a function.

How can you determine the exact point where the function crosses the x-axis?

To determine the exact point where the function crosses the x-axis, you can set the function equal to zero and solve for the x-value. This will give you the coordinate of the point where the function crosses the x-axis.

What does the interval [0,1] represent in this expression?

The interval [0,1] represents the range of values for the x-axis. In this case, it means that the function crosses the x-axis at one point between the values of 0 and 1.

What is the relationship between the function and the x-axis in this expression?

The function and the x-axis are related in that the function crosses the x-axis at one point, meaning that the value of the function is equal to zero at that point. This can also be seen as the x-intercept of the function.

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