Math Problem Statement
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)).
Find the equation of the tangent lines through A and B.
Hint: This requires solving for a, b but as a function of c.
Solution
To solve for the equations of the tangent lines that pass through the origin for the functions and , we need to:
- Find the points and , where the tangent lines of these functions pass through the origin.
- Determine the slopes of the tangent lines at and , then solve for and .
Step 1: Tangent Line and Slope
For any function , the equation of the tangent line at a point can be written as: This tangent line passes through the origin , so we substitute and into the tangent line equation: This simplifies to: or equivalently:
Step 2: Apply to
The derivative of is:
Now, use the condition : Simplifying: Dividing both sides by (which is non-zero): Thus:
Step 3: Apply to
The derivative of is:
Now, use the condition : Simplifying: Dividing both sides by (which is non-zero): Thus:
Step 4: Equation of the Tangent Lines
Now that we have and , we can find the equations of the tangent lines at these points.
- Tangent line at :
The slope of the tangent line at is : Thus, the equation of the tangent line at is: Simplifying:
- Tangent line at :
The slope of the tangent line at is : Thus, the equation of the tangent line at is: Simplifying:
Conclusion
- The equation of the tangent line through is:
- The equation of the tangent line through is:
Would you like further clarifications or details?
Here are 5 related questions:
- How do the slopes of the tangent lines relate to the exponential functions and ?
- What is the geometric interpretation of the points and in terms of symmetry?
- How would the solution change if the functions had different exponential bases?
- Can you extend this method to find tangents for more general exponential functions, such as ?
- What happens to the tangent lines as approaches 0 or infinity?
Tip: Understanding how to compute the slope of tangent lines using derivatives is essential for solving many problems involving curves and tangents in calculus.
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Math Problem Analysis
Mathematical Concepts
Calculus
Exponential Functions
Derivatives
Tangents
Formulas
f(x) = e^{-cx}
g(x) = e^{cx}
f'(x) = -c e^{-cx}
g'(x) = c e^{cx}
Tangent Line Equation: y - f(a) = f'(a)(x - a)
Theorems
Derivative of Exponential Functions
Suitable Grade Level
Undergraduate Calculus
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