Calculus is the branch of mathematics that studies change and accumulation. It gives us tools for describing how quantities vary from one moment to the next and how small pieces add up to a whole. This matters in physics, engineering, economics, biology, and data science because many real systems involve motion, growth, decay, or total effects over time.
The two central ideas of calculus are derivatives and integrals. A derivative measures an instantaneous rate of change, which appears on a graph as the slope of a tangent line. An integral measures accumulation, which appears on a graph as the area under a curve.
These ideas are connected because differentiation and integration undo each other in the relationship called the Fundamental Theorem of Calculus.
Key Facts
- Derivative: f'(x) = lim as h -> 0 of [f(x + h) - f(x)] / h
- A derivative gives the slope of the tangent line to a curve at a point.
- If position is x(t), then velocity is v(t) = dx/dt and acceleration is a(t) = dv/dt.
- Integral: ∫ from a to b f(x) dx gives the signed area under f(x) from x = a to x = b.
- Accumulation can be estimated by rectangles: total ≈ Σ f(x_i) Δx.
- Fundamental Theorem of Calculus: if F'(x) = f(x), then ∫ from a to b f(x) dx = F(b) - F(a).
Vocabulary
- Derivative
- A derivative is the instantaneous rate of change of a function with respect to its input.
- Tangent line
- A tangent line is a line that touches a curve at a point and has the same slope as the curve there.
- Integral
- An integral is a mathematical operation that measures accumulated quantity, often shown as area under a curve.
- Limit
- A limit describes the value a function or expression approaches as the input gets close to a chosen value.
- Fundamental Theorem of Calculus
- The Fundamental Theorem of Calculus states that derivatives and integrals are inverse processes under appropriate conditions.
Common Mistakes to Avoid
- Confusing average rate with instantaneous rate. Average rate uses a secant line over an interval, while instantaneous rate uses a tangent line at one point.
- Treating every integral as ordinary geometric area. A definite integral gives signed area, so parts below the x-axis count as negative.
- Forgetting the dx in an integral. The dx shows the variable of integration and represents the tiny width of each accumulated piece.
- Thinking a derivative only works for straight lines. Derivatives also describe curves by finding the slope of the tangent line at each point.
Practice Questions
- 1 For f(x) = x^2, find the average rate of change from x = 2 to x = 5, then find the instantaneous rate of change at x = 2 using f'(x) = 2x.
- 2 Compute the definite integral ∫ from 0 to 3 2x dx and interpret the answer as an area under a curve.
- 3 A car's velocity graph is positive but decreasing over a time interval. Explain what the derivative of velocity and the integral of velocity tell you about the car's motion.