Math Problem Statement

  1. [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 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?

Solution

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−cxf(x) = e^{-cx}
  • g(x)=ecxg(x) = e^{cx}

where c>0c > 0 is a constant. The points A=(a,f(a))A = (a, f(a)) and B=(b,g(b))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))A = (a, f(a)), and likewise for B=(b,g(b))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)f(x) and g(x)g(x) will give the slopes of the tangent lines.

For f(x)=e−cxf(x) = e^{-cx}, f′(x)=−ce−cxf'(x) = -c e^{-cx} The slope at point A=(a,f(a))A = (a, f(a)) is f′(a)=−ce−caf'(a) = -c e^{-ca}.

For g(x)=ecxg(x) = e^{cx}, g′(x)=cecxg'(x) = c e^{cx} The slope at point B=(b,g(b))B = (b, g(b)) is g′(b)=cecbg'(b) = c e^{cb}.

Step 2: Solve for the points aa and bb

Since the tangent lines pass through the origin, we can use the fact that the line through a point (x1,y1)(x_1, y_1) with slope mm is y=m(x−x1)y = m(x - x_1). Here, the line also passes through (0,0)(0,0).

For f(x)f(x): f(a)=e−ca,and the slope at A is −ce−ca.f(a) = e^{-ca}, \quad \text{and the slope at } A \text{ is } -c e^{-ca}. The equation of the tangent line through AA is: y=−ce−ca(x−a)+e−cay = -c e^{-ca} (x - a) + e^{-ca} Since it passes through the origin (0,0)(0,0), substitute x=0x = 0 and y=0y = 0: 0=−ce−ca(−a)+e−ca0 = -c e^{-ca} (-a) + e^{-ca} 0=cae−ca+e−ca0 = c a e^{-ca} + e^{-ca} 0=e−ca(ca+1)0 = e^{-ca}(c a + 1) This implies: ca+1=0⇒a=−1cc a + 1 = 0 \quad \Rightarrow \quad a = -\frac{1}{c}

For g(x)g(x), using a similar approach: g(b)=ecb,and the slope at B is cecb.g(b) = e^{cb}, \quad \text{and the slope at } B \text{ is } c e^{cb}. The equation of the tangent line through BB is: y=cecb(x−b)+ecby = c e^{cb} (x - b) + e^{cb} Substituting (0,0)(0,0): 0=cecb(−b)+ecb0 = c e^{cb} (-b) + e^{cb} 0=−cbecb+ecb0 = -c b e^{cb} + e^{cb} 0=ecb(1−cb)0 = e^{cb} (1 - c b) This gives: 1−cb=0⇒b=1c1 - c b = 0 \quad \Rightarrow \quad b = \frac{1}{c}

Thus, a=−1ca = -\frac{1}{c} and b=1cb = \frac{1}{c}.

Equation of the tangent lines:

  • At A=(−1c,e1c)A = \left(-\frac{1}{c}, e^{\frac{1}{c}}\right), the tangent line is: y=−e1/c(x+1c)y = -e^{1/c} (x + \frac{1}{c})

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

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Math Problem Analysis

Mathematical Concepts

Calculus
Exponential Functions
Tangent Lines
Orthogonality
Area Under a Curve

Formulas

f'(x) = -c e^(-cx)
g'(x) = c e^(cx)
Equation of tangent line: y = m(x - x1) + y1
Orthogonality condition: vA · vB = 0
Area calculation using definite integrals

Theorems

Orthogonality of Vectors
Fundamental Theorem of Calculus

Suitable Grade Level

College Level (Calculus and Linear Algebra)

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