https://file.aiquickdraw.com/mathbot/file/1726448389644ilaldqgg.jpg

2. [8 points] Let c > 0 be some constant and consider the plots of the functions f(x) = e

−cx and g(x) = e
cx below.
Let A = (a, f(a)) be the point where the tangent line of f goes through the origin and likewise B = (b, g(b)).

(a) [2 points] Find the equation of the tangent lines through A and B.
Hint: This requires solving for a, b but as a function of c.

(b) [2 points] The tangent lines can be written in vector form as
{vAt : t ∈ R} and {vBt : t ∈ R}

What are the vectors vA and vB and for what value of c are these orthogonal (at right angles)?
(c) [4 points] What is the area of the shaded region bounded by the tangent lines and the functions f(x) for
x ≤ 0 and g(x) for x ≥ 0? Provided the answer for arbitrary values of c > 0.

Let's break this problem down step-by-step based on the provided image and the questions.

### Problem Breakdown:
We are given two functions:
- \( f(x) = e^{-cx} \)
- \( g(x) = e^{cx} \)

where \(c > 0\) is a constant. The points \(A = (a, f(a))\) and \(B = (b, g(b))\) represent the points where the tangent lines to these functions pass through the origin.

### (a) Find the equation of the tangent lines at A and B.

To find the equations of the tangent lines, we need:
1. **Slope of the tangent line** at point \(A = (a, f(a))\), and likewise for \(B = (b, g(b))\).
2. **Equation of the tangent line**, using the point-slope form.

#### Step 1: Find the slopes of the tangent lines
The derivative of \(f(x)\) and \(g(x)\) will give the slopes of the tangent lines.

For \( f(x) = e^{-cx} \),
\[
f'(x) = -c e^{-cx}
\]
The slope at point \(A = (a, f(a))\) is \( f'(a) = -c e^{-ca} \).

For \( g(x) = e^{cx} \),
\[
g'(x) = c e^{cx}
\]
The slope at point \(B = (b, g(b))\) is \( g'(b) = c e^{cb} \).

#### Step 2: Solve for the points \(a\) and \(b\)

Since the tangent lines pass through the origin, we can use the fact that the line through a point \((x_1, y_1)\) with slope \(m\) is \(y = m(x - x_1)\). Here, the line also passes through \((0,0)\).

For \(f(x)\):
\[
f(a) = e^{-ca}, \quad \text{and the slope at } A \text{ is } -c e^{-ca}.
\]
The equation of the tangent line through \(A\) is:
\[
y = -c e^{-ca} (x - a) + e^{-ca}
\]
Since it passes through the origin \((0,0)\), substitute \(x = 0\) and \(y = 0\):
\[
0 = -c e^{-ca} (-a) + e^{-ca}
\]
\[
0 = c a e^{-ca} + e^{-ca}
\]
\[
0 = e^{-ca}(c a + 1)
\]
This implies:
\[
c a + 1 = 0 \quad \Rightarrow \quad a = -\frac{1}{c}
\]

For \(g(x)\), using a similar approach:
\[
g(b) = e^{cb}, \quad \text{and the slope at } B \text{ is } c e^{cb}.
\]
The equation of the tangent line through \(B\) is:
\[
y = c e^{cb} (x - b) + e^{cb}
\]
Substituting \((0,0)\):
\[
0 = c e^{cb} (-b) + e^{cb}
\]
\[
0 = -c b e^{cb} + e^{cb}
\]
\[
0 = e^{cb} (1 - c b)
\]
This gives:
\[
1 - c b = 0 \quad \Rightarrow \quad b = \frac{1}{c}
\]

Thus, \(a = -\frac{1}{c}\) and \(b = \frac{1}{c}\).

#### Equation of the tangent lines:
- At \(A = \left(-\frac{1}{c}, e^{\frac{1}{c}}\right)\), the tangent line is:
\[
y = -e^{1/c} (x + \frac{1}{c})
\]

- At \(B = \left(\frac{1}{c}, \right-handSideappend}","\

Let c > 0 be some constant and consider the plots of the functions f(x) = e^(-cx) and g(x) = e^(cx) below. Let A = (a, f(a)) be the point where the tangent line of f goes through the origin and likewise B = (b, g(b)). (a) Find the equation of the tangent lines through A and B. (b) What are the vectors vA and vB and for what value of c are these orthogonal? (c) What is the area of the shaded region bounded by the tangent lines and the functions f(x) for x ≤ 0 and g(x) for x ≥ 0?

Explore the properties of exponential functions, their tangent lines, orthogonality of vectors, and the area bounded by curves. This calculus problem involves finding the equations of tangent lines to f(x) = e^(-cx) and g(x) = e^(cx), solving for orthogonal vectors, and calculating the area of a shaded region for any positive constant c.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation