Math Problem Statement

Find an equation for a line that is tangent to the graph of y = e^x and goes through the origin.

Solution

To solve the problem in Example 5, we aim to find the equation of a tangent line to the curve y=exy = e^x that passes through the origin (0,0)(0,0).

Step-by-step solution:

  1. Determine the slope of the tangent line: The derivative of y=exy = e^x is dydx=ex\frac{dy}{dx} = e^x. This gives the slope of the tangent line at any point xx on the curve.

  2. Write the equation of the tangent line: The general equation of a line is: y=mx+cy = mx + c Here, m=eam = e^a, where aa is the xx-coordinate of the point of tangency, since the slope of the curve at x=ax = a is eae^a. The point of tangency on the curve is (a,ea)(a, e^a).

  3. Condition for the tangent line passing through the origin: If the line passes through the origin (0,0)(0,0), substituting x=0x = 0 and y=0y = 0 into the line equation yields: 0=ea0+c    c=00 = e^a \cdot 0 + c \implies c = 0 Therefore, the line equation simplifies to: y=eaxy = e^a x

  4. Point of tangency constraint: The tangent line must also pass through the point of tangency (a,ea)(a, e^a). Substituting this point into the line equation: ea=eaae^a = e^a \cdot a Simplifying: ea(1a)=0e^a (1 - a) = 0 Since ea0e^a \neq 0, it follows that: 1a=0    a=11 - a = 0 \implies a = 1

  5. Equation of the tangent line: The point of tangency is (1,e1)=(1,e)(1, e^1) = (1, e), and the slope is e1=ee^1 = e. The tangent line is: y=exy = e x

Final Answer:

The equation of the tangent line is: y=exy = e x


Would you like further clarification or additional details? Here are related questions to expand upon:

  1. What is the significance of finding tangents to a curve in real-world applications?
  2. Can a tangent line to y=exy = e^x pass through any other fixed point?
  3. How does the derivative help in determining tangency?
  4. Why is exe^x unique in terms of its derivative and tangent properties?
  5. How would the solution change if the curve was y=e2xy = e^{2x} instead?

Tip: The exponential function exe^x is special because its derivative is the same as the function itself, making its slope at any point straightforward to compute!

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Math Problem Analysis

Mathematical Concepts

Calculus
Exponential Functions
Tangent Lines

Formulas

y = mx + c (Equation of a straight line)
dy/dx = e^x (Derivative of y = e^x)

Theorems

Derivative Rule for Exponential Functions

Suitable Grade Level

Grades 11-12