Calculus studies change, and its two main branches look at change from complementary directions. Differential calculus zooms in on a function to measure an instantaneous rate of change, shown by the slope of a tangent line. Integral calculus adds up many tiny pieces to measure accumulation, shown by the area under a curve.
Together, they let us model motion, growth, energy, probability, and many other changing quantities.
Key Facts
- Derivative definition: f'(x) = lim(h -> 0) [f(x + h) - f(x)] / h
- A derivative gives the instantaneous rate of change and the slope of the tangent line to y = f(x).
- Definite integral: integral from a to b of f(x) dx gives the signed area under f(x) from x = a to x = b.
- Net change theorem: integral from a to b of f'(x) dx = f(b) - f(a).
- Fundamental theorem of calculus: if F'(x) = f(x), then integral from a to b of f(x) dx = F(b) - F(a).
- Units help distinguish the ideas: if f(t) is velocity in m/s, then f'(t) is acceleration in m/s^2 and integral f(t) dt is displacement in m.
Vocabulary
- Derivative
- The derivative of a function is its instantaneous rate of change at a point.
- Tangent line
- A tangent line touches a curve at a point and has the same slope as the curve there.
- Definite integral
- A definite integral represents the signed accumulation of a function over an interval.
- Antiderivative
- An antiderivative of f(x) is a function F(x) whose derivative is f(x).
- Fundamental theorem of calculus
- The fundamental theorem of calculus connects derivatives and integrals by showing that integration can be undone by differentiation.
Common Mistakes to Avoid
- Confusing average rate with instantaneous rate. The slope between two points is an average rate, while the derivative at one point is found by taking the limit as the interval shrinks to zero.
- Treating every area under a curve as positive. A definite integral gives signed area, so regions below the x-axis subtract from the total.
- Forgetting the constant of integration in an indefinite integral. Since many functions can have the same derivative, integral f(x) dx must include + C.
- Using the original function instead of an antiderivative for a definite integral. To compute integral from a to b of f(x) dx, evaluate F(b) - F(a) where F'(x) = f(x).
Practice Questions
- 1 For f(x) = x^2 + 3x, find f'(x) and the slope of the tangent line at x = 2.
- 2 Compute integral from 1 to 4 of 2x dx, and interpret the result as an accumulated quantity.
- 3 A graph of velocity versus time stays above the time axis, increases for a while, then decreases but remains positive. Explain what the derivative of velocity and the integral of velocity represent during this motion.