e^x
Definition
The exponential function e^x represents continuous growth or decay over time. It is defined as raising Euler's number (approximately 2.71828) to the power of x.
Analogy
Think of e^x as compound interest on steroids. Just like money grows exponentially when invested with compound interest over time, e^x shows how quantities can grow or decay rapidly based on continuous multiplication by Euler's number.
Related terms
Exponential Growth/Decay: A process where a quantity increases or decreases by a fixed percentage over equal intervals of time.
Natural Logarithm (ln): The inverse operation of exponentiation with base e. ln(x) gives us the power we need to raise e to obtain x.
Differential Equations: Equations involving derivatives that describe rates of change in various scientific and mathematical contexts.
"e^x" appears in:
Study guides (2)
Practice Questions (19)
- What is the limit of (e^x - 1)/(x) as x approaches 0?
- What is the limit of f(x) = (e^x + e^(-x))/2 as x approaches infinity?
- What is the limit of f(x) = (e^x - 1)/x as x approaches 0?
- Find the limit of f(x) = (e^x - x^2)/(e^x + x^2) as x approaches infinity.
- g(x) = e^x + ln(x). What is the derivative of the function at any point x?
- What is the derivative of f(x) = e^x?
- Given the function f(x) = e^x * (4x^3 + x^2 - 2x), what is the derivative of f(x)?
- What is the derivative of y(x) = e^x / (4x^3 + x)?
- If $p(x) = 2 \cdot \cot^{-1}(e^x)$, then $p'(x)$ is:
- What is the derivative of f(x) = e^x + ln(x)?
- Determine the indefinite integral of 6e^x + 2sin(x).
- Evaluate the indefinite integral: ∫(5e^x - 2cos(x)) dx.
- What is the general solution for the differential equation dy/dx = 4e^x?
- Solve the differential equation: dy/dx = 4e^x, given the initial condition y(0) = 1.
- What is the volume of the solid formed by revolving the region bounded by the functions g(x) = e^x and h(x) = 2e^x around the x-axis from x = 0 to x = 1?
- Find the area between the curves y = e^x and y = x for 0 ≤ x ≤ 1.
- The curve y = e^x is defined over the interval [0, ln(2)]. What is the arc length of the curve?
- What is the Maclaurin series for e^x?
- Which of the following is the first term of the Maclaurin series of $6xe^x$?
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