Calculus Explorer: Limits, Derivatives and Integrals

Learning objectives

• Estimate one-sided and two-sided limits, and explain why a limit can differ from a function’s value.
• Recognise continuous functions and removable, jump and infinite discontinuities.
• Interpret the derivative as the limit of secant slopes and the slope of the tangent line.
• Use the sign of f′ to find where f increases or decreases, and classify stationary points correctly.
• Approximate definite integrals with left, right and midpoint Riemann sums, and distinguish signed area from geometric area.
• Explain the fundamental theorem of calculus.

Limits

The limit of f(x) as x approaches a describes the values f(x) gets close to when x is close to a, but not equal to it. We write lim_x→a f(x) = L. The value f(a) itself does not matter: f might not even be defined at a.

A left-hand limit (x → a⁻) uses only x < a; a right-hand limit (x → a⁺) uses only x > a. The two-sided limit exists only when both one-sided limits exist and are equal. If f grows without bound, we write lim = ∞ to describe the behaviour, but the limit does not exist as a number.

Continuity and discontinuities

f is continuous at a when three things hold: f(a) is defined, the limit exists, and the limit equals f(a).

Derivatives

The slope of a secant line through (x, f(x)) and (x + h, f(x + h)) is the average rate of change [f(x + h) − f(x)]/h. Letting h shrink to zero gives the derivative, theinstantaneous rate of change:

f′(x) = lim_h→0 [f(x + h) − f(x)] / h

Geometrically, f′(x) is the slope of the tangent line, the line that just touches the curve there. A function is differentiable at x when this limit exists. Corners (|x| at 0), cusps (x^2/3 at 0), jumps and vertical tangents have no derivative. A differentiable function is always continuous, but a continuous function need not be differentiable.

What f′ tells you about f

• Where f′(x) > 0, f is increasing; where f′(x) < 0, f is decreasing.
• Where f′(x) = 0, the tangent is horizontal: a stationary point.
• A stationary point is a local maximum if f′ changes from + to −, a local minimum if it changes from − to +, and neither if f′ keeps its sign (x³ at 0).
• Maxima and minima can also occur where f′ does not exist, such as the corner of |x|.

Integrals and Riemann sums

To find the area under a curve between x = a and x = b, split [a, b] into n strips of width Δx = (b − a)/n, and add up rectangles of height f(xᵢ*), where xᵢ* is the left end, right end or midpoint of each strip. This is aRiemann sum, Σ f(xᵢ*)·Δx. As n → ∞ the sums approach the definite integral∫_a^b f(x) dx.

An indefinite integral ∫ f(x) dx = F(x) + C is the family of all antiderivatives(functions whose derivative is f). A definite integral is a number; an indefinite integral is a family of functions. When F is known, ∫_a^b f(x) dx = F(b) − F(a).

For an increasing function, left sums underestimate and right sums overestimate; midpoint sums are usually much more accurate. The error of left and right sums shrinks roughly in proportion to 1/n, and of midpoint sums to 1/n².

Signed area versus geometric area

The definite integral measures signed accumulation: area above the x-axis counts positive and area below counts negative. So ∫₀^2π sin x dx = 0, even though the curve encloses a totalgeometric area of ∫₀^2π |sin x| dx = 4. If f is unbounded on [a, b] (for example 1/x across 0), the ordinary integral does not exist; such “improper” integrals need separate treatment.

The fundamental theorem of calculus

Define the accumulation function A(x) = ∫_a^x f(t) dt: the signed area from a up to a moving edge x. If f is continuous on an interval containing a and x, then

d/dx [ ∫_a^x f(t) dt ] = f(x)

In words: the area grows at a rate equal to the height of the curve at the moving edge. When f is positive A increases, when f is negative A decreases, and where f crosses zero A has a peak or a valley. The second part of the theorem gives ∫_a^b f(x) dx = F(b) − F(a) for any antiderivative F, which is how exact integrals are calculated.

Formula reference

How the lab computes

A computer cannot take a true limit, so the lab estimates. Limits are estimated from f(a ± 10⁻ᵏ) for k up to 7 and labelled “numerical”. Derivatives of the preset functions use their exact formulas; for your own functions the lab uses the central difference [f(x + h) − f(x − h)]/(2h) with a small h, which is accurate to about eight significant digits for smooth functions. Corners and cusps are detected by comparing the slopes from the left and the right. Exact integrals come from an antiderivative when one is known; otherwise the lab uses two-point Gauss–Legendre quadrature with 1,000 panels. The lab flags intervals where f is unbounded instead of printing a meaningless number. Expressions you type are read by a small parser that only understands maths, never by running code. Very rapidly oscillating or extremely steep functions can still fool any numerical method.

Real-world applications

• Physics: velocity is the derivative of position, acceleration is the derivative of velocity, and distance travelled is the integral of speed.
• Engineering: integrals give the work done by a force, the centre of mass of a beam and the charge stored in a capacitor.
• Economics: marginal cost and marginal revenue are derivatives; total cost is the integral of marginal cost.
• Medicine: the area under a drug-concentration curve measures total exposure to a medicine.
• Computing: machine learning trains models by following derivatives (gradients) downhill.

Interesting facts

• Isaac Newton and Gottfried Leibniz developed calculus independently in the 1660s–1680s; the notation dy/dx and ∫ is Leibniz’s.
• Archimedes estimated the area of a parabolic segment more than 2,000 years ago by filling it with ever-smaller triangles, an early version of a limit.
• The rigorous ε–δ definition of a limit was only completed in the 19th century, by Cauchy and Weierstrass.
• Weierstrass found a function that is continuous everywhere but differentiable nowhere: a curve made entirely of “corners”.

Common misconceptions

• “The limit is just f(a).” Only for continuous functions. A limit can exist where f(a) is undefined or different.
• “f′(a) = 0 means a maximum or minimum.” Not always: x³ has f′(0) = 0 but no turning point.
• “Continuous means differentiable.” |x| is continuous at 0 but has no derivative there.
• “An integral is always the area.” It is signed area: parts below the axis subtract.
• “The secant slope is the derivative.” The secant slope is an average rate of change; the derivative is its limit.

Important terms

Limit

The value f(x) approaches as x approaches a, from one side or both.

Continuity

f is continuous at a if f(a) exists and equals the limit there.

Derivative

The instantaneous rate of change; the slope of the tangent line.

Secant line

A line through two points of a curve; its slope is an average rate of change.

Tangent line

The line through a point of a curve with slope equal to the derivative there.

Differentiable

Having a derivative at a point (no corner, cusp, jump or vertical tangent).

Stationary point

A point where f′ = 0.

Riemann sum

A sum of rectangle areas that approximates an integral.

Definite integral

The limit of Riemann sums on [a, b]: a number giving signed accumulation.

Indefinite integral (antiderivative)

A family of functions F(x) + C whose derivative is f.

Key takeaways

• A limit describes behaviour near a point, not the value at it.
• The derivative is the limit of secant slopes: the slope of the tangent.
• The sign of f′ shows where f rises and falls; turning points need f′ to change sign.
• Integrals are limits of Riemann sums and measure signed area.
• The fundamental theorem links the two ideas: accumulating f and then differentiating gives f back.