Difficulty: Introductory | Prerequisites: Familiarity with functions, basic algebra, and the coordinate plane.
Differential calculus is the branch of mathematics concerned with how quantities change. It gives you the tools to describe rates of change (velocity, acceleration) and the geometry of curves (slope, arc length). This material sits at the very start of MATH 104 and underpins nearly everything that follows, from optimisation problems to integral calculus. If you are comfortable with functions, graphs, and the idea that a line has a steepness you can measure, you have what you need to begin.
Differential calculus studies how things change, using the concept of the derivative to measure instantaneous rates of change. The foundational ideas, including slope, velocity, acceleration, optimisation, and arc length, all connect back to one question: "How fast is this quantity changing right now?"
Hyperreal numbers
An extension of the real numbers (denoted *R) that includes infinitely large and infinitely small quantities. Introduced by Edwin Hewitt in 1948, they contain numbers greater than any finite sum 1 + 1 + ... + 1, and reciprocals of such numbers are infinitesimals.
In simple terms, hyperreals are a formal way of saying "bigger than any real number" and "smaller than any positive real number but not zero." They let you build calculus using Leibniz-style reasoning about infinitely small changes.
Velocity
The rate of change of an object's position with respect to a frame of reference, expressed as a function of time. It specifies both speed and direction (e.g. 60 km/h to the north).
Think of it as: speed with a direction attached. Differential calculus computes velocity by taking the derivative of a position function.
Acceleration
The rate of change of velocity of an object. It is the net result of all forces acting on the object, as described by Newton's Second Law.
In simple terms, acceleration tells you how quickly the velocity itself is changing. It is the derivative of velocity, or equivalently the second derivative of position.
Slope
The gradient of a line, a number that describes both its direction and steepness, usually denoted by m. Calculated as the ratio of vertical change to horizontal change between any two distinct points on the line ("rise over run").
Think of it as: how many units up (or down) you move for every unit you move to the right. In differential calculus, the slope of a curve at a point is the value of the derivative at that point.
Mathematical optimisation
The process of selecting a best element from a set of available alternatives, typically by maximising or minimising a real function.
In simple terms, this is using calculus to find the highest or lowest value a function can take. Setting the derivative equal to zero is the standard first step.
Arc length
The distance measured along a curve between two points. Determining the arc length of an irregular curve is called rectification. Integral calculus provides a general formula for computing it.
Think of it as: if you could straighten out a piece of curved wire, how long would it be?
Slope (m) = rise / run = (y₂ - y₁) / (x₂ - x₁)
A positive slope (m > 0) means the line rises from left to right.
A negative slope (m < 0) means the line falls from left to right.
A zero slope (m = 0) means a horizontal line, which represents a constant function.
An undefined slope occurs when the line is vertical (division by zero in the formula).
The steepness of a line is measured by the absolute value of its slope. A larger |m| means a steeper line.
Differential calculus extends this idea from straight lines to curves: the slope of a curve at any point is the derivative evaluated at that point.
Position is a function of time: s(t).
Velocity is the first derivative of position: v(t) = s'(t). It carries both magnitude (speed) and direction.
Acceleration is the first derivative of velocity, or the second derivative of position: a(t) = v'(t) = s''(t).
Newton's Second Law connects acceleration to force: F = ma. This is one of the most direct physical applications of the derivative.
An optimisation problem asks: at what input does a function reach its maximum or minimum value?
The standard approach is to find critical points by setting the first derivative equal to zero and then testing with the second derivative or a sign chart.
This appears throughout MATH 104, from simple max/min word problems to applied problems in physics and economics.
Arc length measures the distance along a curve, as distinct from the straight-line distance between two endpoints.
Computing arc length historically required curve-specific methods. Infinitesimal calculus provides a general integral formula.
Rectification is the process of finding a straight segment equal in length to a given arc.
This topic bridges differential and integral calculus: the integrand involves the derivative of the function defining the curve.
Slope of a line
m = \frac{y_2 - y_1}{x_2 - x_1}Velocity (first derivative of position)
v(t) = \frac{ds}{dt} = s'(t)Acceleration (second derivative of position)
a(t) = \frac{dv}{dt} = \frac{d^2 s}{dt^2} = s''(t)Arc length of a curve y = f(x) from x = a to x = b
L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dxVelocity and acceleration are how physicists and engineers describe motion. Every time your phone's GPS computes your speed or a car's anti-lock braking system adjusts wheel pressure, derivatives are doing the work behind the scenes.
Slope is the foundation of linear regression in statistics: fitting a line to data points means finding the slope that minimises prediction error.
Optimisation is used constantly in economics (maximising profit, minimising cost), engineering (finding the dimensions that use the least material), and machine learning (gradient descent is an optimisation algorithm at its core).
Arc length matters in road design, cable routing, and any situation where you need to know the true distance along a curved path rather than the straight-line shortcut.
Students often treat speed and velocity as interchangeable. Speed is the magnitude only; velocity includes direction. A car doing 60 km/h east and one doing 60 km/h west have the same speed but different velocities.
A zero slope does not mean "no line." It means the line is perfectly horizontal, a constant function.
Students sometimes assume that an undefined slope means the slope is zero. A vertical line has an undefined slope because you would be dividing by zero in the rise-over-run formula. This is a different situation from a horizontal line.
In optimisation, finding where the derivative equals zero is necessary but not sufficient. You still need to confirm whether each critical point is a maximum, a minimum, or neither (a saddle point or inflection point).
⚠️ Slope is the conceptual backbone of the derivative. Expect questions that ask you to connect rise-over-run to the limit definition of a derivative.
⚠️ Velocity and acceleration problems are standard fare on MATH 104 exams. Know how to move between position, velocity, and acceleration by differentiating (and, later, integrating).
⚠️ Optimisation problems often appear as multi-step word problems. The derivative-equals-zero step is only the middle of the solution; setting up the function and verifying the nature of the critical point are where marks are won or lost.
⚠️ The arc length formula is frequently tested as a "set up the integral" problem. You may not need to evaluate it, but you do need to write it correctly.
True or false: A vertical line has a slope of zero. (False. A vertical line has an undefined slope.)
Fill in the blank: Velocity is the ______ derivative of the position function. (First.)
True or false: Acceleration is the rate of change of position. (False. Acceleration is the rate of change of velocity.)
Fill in the blank: In the slope formula, "rise over run" means ______ change divided by ______ change. (Vertical; horizontal.)
True or false: A critical point where the derivative is zero is always a maximum or minimum. (False. It could be an inflection point or saddle point.)
Q: A particle's position is given by s(t) = 3t² + 2t. What is its velocity at t = 4?
A: v(t) = s'(t) = 6t + 2. At t = 4, v(4) = 6(4) + 2 = 26.
Q: What is the slope of the line passing through the points (1, 3) and (5, 11)?
A: m = (11 - 3) / (5 - 1) = 8 / 4 = 2.
Q: Explain why a vertical line does not have a slope.
A: For a vertical line, all points share the same x-coordinate. The denominator in rise/run becomes zero, and division by zero is undefined.
Q: A function f(x) = x³ - 12x has critical points. Find them and determine whether each is a local max or min.
A: f'(x) = 3x² - 12. Setting f'(x) = 0 gives x² = 4, so x = 2 and x = -2. f''(x) = 6x. At x = 2, f''(2) = 12 > 0, so x = 2 is a local minimum. At x = -2, f''(-2) = -12 < 0, so x = -2 is a local maximum.
Q: Write the arc length integral for the curve y = x² from x = 0 to x = 3.
A: dy/dx = 2x. The arc length is L = ∫₀³ √(1 + (2x)²) dx = ∫₀³ √(1 + 4x²) dx.
Slope leads directly into the formal limit definition of the derivative, which is the centrepiece of MATH 104. Once you are comfortable with slope as rise-over-run, the derivative is simply the limit of that ratio as the two points get infinitely close together.
Velocity and acceleration reappear in integral calculus: if differentiation turns position into velocity, integration turns velocity back into position. Many later problems will ask you to move in both directions.
Optimisation connects to curve sketching (finding maxima, minima, and inflection points), which is one of the major applications sections later in the course.
Differential calculus, MATH 104, Calculus 1, Penn State, derivative, slope, gradient, rise over run, velocity, speed, acceleration, Newton's Second Law, rate of change, instantaneous rate of change, hyperreal numbers, nonstandard reals, infinitesimal, mathematical optimisation, optimization, max and min, critical points, arc length, rectification, curve length, integral calculus, limit definition of the derivative