Find the area of the triangle formed by the x -axis and the tangent lines to the curves y=e^a x

Find the area of the triangle formed by the x -axis and the...

Key Concepts

Exponential Functions

Exponential functions are characterized by their constant rates of growth or decay, and in this context they are given in the form e^(ax) and e^(-ax). These functions have important properties, such as e^(0) being equal to 1, which simplifies the evaluation of points on the curve. Understanding the behavior and symmetry of exponential functions helps in accurately determining the specific points and slopes required for forming tangent lines.

Intersection Points

The process of finding intersection points involves determining where lines or curves meet. In problems like this, it is important to find where the tangent lines intersect the x-axis. This typically involves setting the y-value of the line equal to zero and solving for x. These intersection points are crucial as they serve as vertices for geometric figure constructions, such as triangles.

Triangle Area Calculation

Once the vertices of the triangle are known, standard geometric principles are employed to calculate the area of the triangle. A common method is using the formula for the area of a triangle, which is half the product of its base and height. This concept integrates knowledge of geometry with the algebraic determination of lengths from calculated points, thereby linking the analytic and geometric viewpoints.

Derivative and Tangent Lines

This concept involves using calculus to compute the slope of a function at a given point by differentiating the function. The slope obtained through the derivative is used to form the equation of a tangent line at that point, typically using the point-slope form. This approach is essential for finding the tangent line to any curve, which in many problems is crucial for analyzing local linear approximations or intersections with other lines or axes.

Transcript

00:01 Okay, first things first for this problem, we need to figure out what the graphs look like in question, and so what the triangle looks like that we're going to try to figure out the area of.

00:10 Well, first off, let's make this, here's our x -axis and here's our y -axis.

00:18 E to the a x.

00:21 Y equals e to the a x.

00:24 It's an exponential function with some constant a in front.

00:28 And then the other graph we're going to consider is e to the negative a x.

00:31 So in the words, whatever a is it's the negative a x.

00:33 Version of that.

00:35 So if a is positive then this function on the left is going to be an increasing exponential and this one's on the right it's going to be a decreasing exponential.

00:42 And vice versa.

00:44 So in the words one of our exponentials is going to be increasing, it's going to look something like this, and one of them is going to be decreasing.

00:52 In fact, in a sort of a mirrored way.

00:54 It's mirrored across the y -axis because that's the transformation.

00:59 Putting a minus sign does that.

01:01 Now the tangent lines are going to look something like this.

01:03 So the tangent line to this say to the first curve is going to look like this.

01:07 Obviously it should be touching, so let's try to get that.

01:10 And then this one should be, well, you can see it should be symmetric.

01:15 So maybe i'll redraw this.

01:21 There we go.

01:22 So you can see there are two tangent lines at x equals zero for these two curves.

01:28 And so the triangles formed this triangle right here.

01:35 Okay, so now how do we figure out the area of this triangle? well, notice that its height, it's, remember area of a triangle, is 1 half times the base length, so that's this length there, times its height.

01:51 Well, actually, the height, if we know that is the y -coordinate of this point.

01:58 Now, we know that when we substitute x equals 0 into any exponential, we get 1.

02:02 So in other words, this y -coordinate is actually 1.

02:05 So the height's 1.

02:06 So in other words, the base is just, the formula is just this...

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