Difficulty: Introductory | Prerequisites: Solid algebra skills (factoring, solving equations, basic graphing). No prior calculus needed.
This overview introduces the three pillars of calculus: functions, derivatives and integrals. Calculus is the mathematics of change and variation, extending algebra from static problems to dynamic ones involving motion, rates of change, optimum values and accumulated totals. Everything in the course builds on these three ideas, viewed from four angles (physical, geometric, numerical, algebraic). If you can hold this bird's-eye map in your head, individual topics will slot into place as you encounter them.
Calculus studies how things change. Derivatives measure the instantaneous rate of change of a function (the slope of a curve, the velocity of a moving object). Integrals measure the accumulated total of a changing quantity (the area under a curve, the total distance travelled). The Fundamental Theorem of Calculus connects the two: integration reverses differentiation.
Function
A rule f : X → Y that assigns each element of an input set (domain) X to exactly one element of an output set Y. In plain terms, a function is a machine: you feed it an input and it spits out one output.
Domain
The set of all allowed inputs for a function. Think of it as the collection of values you are permitted to plug in.
Independent variable (argument)
The input quantity, often written x or t. You choose it freely within the domain.
Dependent variable (value)
The output quantity, determined by the function once the input is fixed.
Derivative
The instantaneous rate of change of a function, written f'(x). In simple terms, it tells you how fast the output is changing at a single moment. Geometrically, it is the slope of the tangent line to the graph.
Integral
The accumulated total of a rate of change over an interval, written ∫ₐᵇ g(x) dx. Think of it as adding up many tiny contributions to get a grand total. Geometrically, it is the area under a curve.
Fundamental Theorem of Calculus
The theorem connecting derivatives and integrals: if g(x) = f'(x), then ∫ₐᵇ g(x) dx = f(b) − f(a). In simple terms, integration and differentiation are reverse operations.
Antiderivative
A function whose derivative gives you the integrand. If f'(x) = g(x), then f(x) is an antiderivative of g(x).
Calculus concepts have four levels of meaning. Every problem starts on one level (usually physical or geometric), gets translated to another level (numerical or algebraic) where it can be solved, then the answer is translated back.
Physical level
A function describes how one real-world quantity determines another. The input is the independent variable; the output is the dependent variable.
Example: a stone dropped from a bridge. Elapsed time t (seconds) determines the fallen distance s (feet). The function is s = f(t), and the domain is 0 ≤ t ≤ 5 (since the stone hits the water at t = 5).
Geometric level
A function is a graph in the coordinate plane: the curve of points (x, y) where y = f(x).
The falling-stone graph curves upward from (0, 0) to (5, 400), getting steeper as the stone accelerates. It is a segment of a parabola.
Numerical level
A function is a table of input-output pairs. In practice you measure sample values and imagine a complete table filling the gaps.
Sample data for the falling stone: t = 0, 1, 2, 3, 4, 5 gives s = 0, 16, 64, 144, 256, 400.
Algebraic level
A function is a formula. The formula lets you compute the output exactly from the input.
The model for the falling stone is f(t) = 16t². This is an approximation of reality (air resistance adds some error).
The derivative f'(x) captures how fast a function's output changes as its input changes.
Physical level
The derivative is the rate of change. For the falling stone, the derivative of position with respect to time is velocity: v = f'(t).
Geometric level
The derivative f'(a) at x = a is the slope of the tangent line to the graph at the point (a, f(a)). The tangent line equation is y = f(a) + f'(a)(x − a).
Numerical level
Approximate the derivative using the difference quotient:
f'(a) ≈ Δf / Δx = [f(a + h) − f(a)] / h
For f(t) = 16t², the approximate velocity at t = 3 with h = 0.1 is [f(3.1) − f(3)] / 0.1 = [153.76 − 144] / 0.1 = 97.6 ft/sec.
Once you know f'(a), you can approximate f(x) near a with a linear function: f(x) ≈ f(a) + f'(a)(x − a). This is called linear approximation.
Algebraic level
The precise definition: the derivative is the limit of the difference quotient as h → 0.
f'(a) = lim(h→0) [f(a + h) − f(a)] / h
Basic derivative rules include the power rule (xⁿ)' = nxⁿ⁻¹ and (sin x)' = cos x. These combine via the Sum, Difference, Constant Multiple, Product, Quotient and Chain Rules.
For f(t) = 16t², the derivative is f'(t) = 32t. At t = 3, the exact velocity is f'(3) = 96 ft/sec.
The integral ∫ₐᵇ g(x) dx accumulates tiny changes over an interval to produce a total.
Physical level
If a quantity z = f(x) changes at rate g(x) = f'(x), the total change from x = a to x = b is the integral: f(b) − f(a) = ∫ₐᵇ g(x) dx.
For the falling stone, knowing velocity v = 32t, the total distance fallen from t = 0 to t = 3 is ∫₀³ 32t dt.
Geometric level
When g(x) ≥ 0, the integral is the area under the curve y = g(x) and above the x-axis, from x = a to x = b. Each thin slice has height g(x) and width Δx, and the area is the cumulative total of those slices.
Numerical level
Approximate the integral by splitting [a, b] into n small increments of width Δx = (b − a)/n, sampling g(xᵢ) in each, and summing:
∫ₐᵇ g(x) dx ≈ g(x₁)Δx + g(x₂)Δx + … + g(xₙ)Δx
For v = 32t with n = 3 and Δt = 1: ∫₀³ 32t dt ≈ 32(1)(1) + 32(2)(1) + 32(3)(1) = 192. This is an overestimate (right-endpoint sampling picks the fastest velocity in each interval). More subintervals give better accuracy.
Algebraic level (Fundamental Theorem of Calculus)
Integrals are reverse derivatives. If g(x) = f'(x) for a known f(x), then:
∫ₐᵇ g(x) dx = f(b) − f(a)
Since f(t) = 16t² has f'(t) = 32t, the exact distance is ∫₀³ 32t dt = 16(3²) − 16(0²) = 144 ft.
Difference quotient (numerical derivative)
f'(a) ≈ [f(a + h) − f(a)] / h
Derivative (limit definition)
f'(a) = lim(h→0) [f(a + h) − f(a)] / h
Power rule
(xⁿ)' = nxⁿ⁻¹
Riemann sum (numerical integral)
∫ₐᵇ g(x) dx ≈ Σ g(xᵢ) Δx, where Δx = (b − a) / n
Fundamental Theorem of Calculus
If g(x) = f'(x), then ∫ₐᵇ g(x) dx = f(b) − f(a)
Tangent line (point-slope form)
y = f(a) + f'(a)(x − a)
Linear approximation
f(x) ≈ f(a) + f'(a)(x − a), for x near a
The falling-stone model f(t) = 16t² is a simplified version of what physicists and engineers use when analysing projectile motion, structural loads and anything where gravity acts on an object. Derivatives underpin every speedometer reading: velocity is the derivative of position. Integrals are how engineers calculate total quantities from rates, from fluid flow through a pipe to the work done by a varying force.
Students often confuse average velocity with instantaneous velocity. Average velocity uses two points on the graph (a secant line); instantaneous velocity is the limit as those points merge (the tangent line).
The numerical approximation (97.6 ft/sec) is close to the exact derivative (96 ft/sec) but is not the same thing. The derivative is the limiting value, not any single approximation.
The Riemann sum overestimate of 192 versus the exact integral of 144 surprises students. Which endpoint you sample matters. Right endpoints on an increasing function always overestimate.
The integral is not "the opposite of the derivative" in a vague sense. It is precisely the reverse operation: if differentiating f gives g, then integrating g recovers f (up to a constant).
⚠️ Be able to state the four levels (physical, geometric, numerical, algebraic) for each concept. Exams frequently ask you to interpret a derivative or integral in context, not just compute one.
⚠️ The Fundamental Theorem of Calculus appears throughout the course. Know the statement: if g = f', then ∫ₐᵇ g(x) dx = f(b) − f(a).
⚠️ You will need the difference quotient to define the derivative from scratch. Make sure you can set it up for any given function.
True or False: The derivative of a function at a point gives the slope of the secant line at that point.
False. It gives the slope of the tangent line. The secant line uses two points; the tangent line is the limit as those points merge.
Fill in the blank: The Fundamental Theorem of Calculus states that if g(x) = f'(x), then ∫ₐᵇ g(x) dx = ______.
f(b) − f(a)
True or False: A Riemann sum with right endpoints always overestimates the integral of an increasing function.
True.
Fill in the blank: The four levels of meaning in calculus are physical, geometric, ______ and algebraic.
Numerical.
True or False: The derivative f'(3) = 96 means the stone is travelling exactly 96 ft/sec at the instant t = 3.
True.
Q: Given f(t) = 16t², compute the average velocity between t = 2 and t = 5.
A: v_avg = [f(5) − f(2)] / (5 − 2) = [400 − 64] / 3 = 336 / 3 = 112 ft/sec.
Q: Using the limit definition of the derivative, find f'(t) for f(t) = 16t².
A: f'(t) = lim(h→0) [16(t+h)² − 16t²] / h = lim(h→0) [16(2th + h²)] / h = lim(h→0) 16(2t + h) = 32t.
Q: What is the geometric interpretation of ∫₀³ 32t dt?
A: It is the area under the line v = 32t and above the t-axis, from t = 0 to t = 3. This area is a triangle with base 3 and height 96, so the area is (1/2)(3)(96) = 144.
Q: A function has f(2) = 10 and f'(2) = 3. Approximate f(2.1) using linear approximation.
A: f(2.1) ≈ f(2) + f'(2)(2.1 − 2) = 10 + 3(0.1) = 10.3.
Q: Why does the numerical approximation 97.6 differ from the exact derivative value 96?
A: Because the numerical approximation uses h = 0.1 (a finite step), not h → 0. There is a small error proportional to h. Only the limit gives the exact value.
This overview maps directly onto the rest of the course. Chapters 1.4 through 1.8 develop the precise notion of a limit, which is the foundation for the derivative definition. Chapters 2 and 3 build the derivative rules (power, product, quotient, chain). Chapter 4 applies derivatives to optimisation and curve sketching. Chapters 5 and 6 develop the integral via Riemann sums and the Fundamental Theorem.
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