Difficulty: Introductory
Prerequisites: Basic algebra (solving linear equations, plotting points on a coordinate plane).
This is where calculus begins, and it starts with something you already understand: if you drive at a steady speed, you can work out how far you have gone. Section 1.1 introduces the idea that calculus studies pairs of functions, not single ones. The first pair is distance f(t) and velocity v(t). If you can read a graph of one, you can figure out the other. This relationship between a function and its slope is the seed of everything that follows in the course.
Calculus connects two functions at once. The opening example is distance and velocity: velocity is the slope of the distance graph, and distance is the area under the velocity graph. When velocity is constant, both graphs are straight lines. When velocity changes in jumps, the distance graph becomes piecewise linear.
Function
A rule that assigns exactly one output to each input. Written as f(t) or y = f(x). In simple terms, you put a number in and get a number out.
Linear function
A function whose graph is a straight line. General form: y = mx + b, where m is the slope and b is the y-intercept. Think of it as the simplest possible relationship between two quantities: one goes up by a fixed amount every time the other goes up by one.
Slope
The rate at which a line rises or falls. Calculated as (change in y) / (change in x). A positive slope means the line goes uphill; a negative slope means it goes downhill; zero slope means it is flat.
Velocity, v(t)
The rate of change of distance with respect to time. It is the slope of the distance graph. In simple terms, velocity tells you how fast the distance is changing at any moment.
Distance function, f(t)
The total distance travelled (or position) as a function of time. It is the area under the velocity graph. Think of it as the running total of how far you have gone.
Piecewise linear function
A function built from several straight-line segments joined end to end. Each segment has its own slope. This is what the distance graph looks like when velocity changes in jumps.
Piecewise constant function
A function that stays flat at one value, then jumps to another value, and stays flat again. This is what the velocity graph looks like when speed changes abruptly (a step function).
Domain
The set of all allowed inputs to a function. For a distance function over a journey, this is the time interval.
Range
The set of all possible outputs of a function. For a distance function, this is the interval of distances actually reached.
Every straight line can be written as y = mx + b, where m is the slope and b is the starting value (the y-intercept).
Equivalent forms appear with different letters: f = vt + C, or y - 4 = 7(x - 2) for a line through (2, 4) with slope 7.
To find the equation of a line through two points (0, b) and (a, 0): the slope is -b/a, and the equation is y = b - (b/a)x.
Reading a graph means extracting the slope and intercept. Writing a graph means translating the equation into a picture. Both skills matter equally.
Calculus studies pairs of functions. The first pair is distance f(t) and velocity v(t).
When velocity is constant: if you travel 120 miles in 2 hours, v = 60 mph. The distance graph is a straight line with slope 60, so f(t) = 60t.
Velocity is the slope of the distance graph. Where f(t) is steepest, v(t) is largest. Where f(t) is flat, v(t) = 0. Where f(t) turns downward, v(t) is negative.
Distance is the area under the velocity graph. Area above the time axis counts as positive (forward travel). Area below counts as negative (backward travel).
When velocity changes in jumps, the velocity graph is piecewise constant (flat segments at different heights).
The corresponding distance graph is piecewise linear (straight segments with different slopes, joined at the breakpoints).
At each breakpoint, the distance is continuous (no gap), but the slope changes suddenly.
To write the formula for each segment, use f(t) = f(a) + v(t - a), where a is the start of that segment, f(a) is the distance at that start, and v is the velocity during that segment.
Flat segment of f(t): velocity is zero (the object is stationary).
Rising segment: velocity is positive (moving forward). Steeper rise means higher velocity.
Falling segment: velocity is negative (moving backward).
The velocity graph is drawn by reading the slope of each segment of the distance graph and plotting that slope as a horizontal line over the same time interval.
y = mx + bSlope-intercept form of a straight line. m = slope, b = y-intercept.
f(t) = f(a) + v(t - a)Point-slope form for distance. Gives the distance at time t when you know the distance at time a and the constant velocity v over that segment.
v = \frac{\Delta f}{\Delta t} = \frac{f(t_2) - f(t_1)}{t_2 - t_1}Velocity equals the slope of the distance graph: the change in distance divided by the change in time.
f = v \times tDistance equals velocity times time (constant velocity, starting from f(0) = 0). This is also the area of the rectangle under the velocity graph.
For a piecewise constant velocity with breakpoints, the complete distance function is built segment by segment. For example, if v = 10 for 0 to 2, v = 0 for 2 to 3, v = 20 for 3 to 4, and v = -5 for 4 to 6:
f(t) = 10t for 0 <= t <= 2
f(t) = 20 for 2 <= t <= 3
f(t) = 20 + 20(t - 3) for 3 <= t <= 4
f(t) = 40 - 5(t - 4) for 4 <= t <= 6
The velocity-distance relationship is how your car's speedometer and odometer work together. The speedometer reads v(t); the odometer accumulates f(t). Cruise control holds v constant, making f a straight line. Every GPS navigation estimate ("arrive in 45 minutes") is a version of f = vt.
Students often confuse the distance graph with the velocity graph. A flat distance graph does not mean the object is moving at constant speed; it means the object is stationary (v = 0).
Negative velocity does not mean "slowing down." It means the object is moving backward. Slowing down means the magnitude of velocity is decreasing.
When velocity jumps at a breakpoint, the distance graph has a corner but no gap. Students sometimes draw a gap in f(t) at the breakpoint, which is wrong: distance is always continuous.
Area below the time axis on the velocity graph counts as negative distance (backward travel), not zero. Forgetting the sign of the area is a common source of error.
⚠️ You will be asked to draw v(t) from a given f(t) graph, and vice versa. Practise reading slopes from graphs until it is automatic.
⚠️ Writing the piecewise formula for f(t) using the point-slope form f(t) = f(a) + v(t - a) is a standard exam question. Get comfortable choosing the right starting point (a, f(a)) for each segment.
⚠️ The connection "velocity = slope of f" and "distance = area under v" appears throughout the entire course. It is the preview of derivatives and integrals.
True or False: If the distance graph f(t) is a straight line with slope 10, then the velocity is v(t) = 10. True.
True or False: If velocity is negative, the distance graph slopes downward. True.
Fill in the blank: If a car travels 80 miles in 4 hours at constant speed, then v = ______ mph. 20.
True or False: At a point where the distance graph is flat, the velocity is at its maximum. False. The velocity is zero.
Fill in the blank: The area under the velocity graph from t = 0 to t = T gives the ______ at time T. Distance (or total displacement).
Q: A car's distance graph shows f(t) = 10t for 0 <= t <= 2, f(t) = 20 for 2 <= t <= 3, and f(t) = 20 + 20(t - 3) for 3 <= t <= 4. What is the velocity during each interval?
A: v = 10 mph for 0 < t < 2, v = 0 for 2 < t < 3, and v = 20 mph for 3 < t < 4.
Q: Suppose v(t) = 30 mph from t = 0 to t = 2, and v(t) = -20 mph from t = 3 to t = 6. If f(0) = 0, what is f(2)? What is f(5)?
A: f(2) = 30 x 2 = 60 miles. From t = 2 to t = 3 velocity is not specified, so assume it requires further information. From t = 3 to t = 6 the car goes backward: f(5) = f(3) + (-20)(5 - 3) = f(3) - 40. You need to know f(3) to complete the answer.
Q: Find the equation of the line through (0, b) and (a, 0).
A: Slope = (0 - b)/(a - 0) = -b/a. Equation: y = b - (b/a)x.
Q: A velocity graph is a step function: v = 3 for 0 < t < 4 and v = 6 for 4 < t < 7, with f(0) = 0. Write the formula for f(t) and find f(7).
A: f(t) = 3t for 0 <= t <= 4. At t = 4, f(4) = 12. Then f(t) = 12 + 6(t - 4) for 4 <= t <= 7. So f(7) = 12 + 6(3) = 30.
Q: On a distance graph, the curve rises steeply, then flattens, then falls. Describe the velocity at the steepest point, the flat point, and the falling part.
A: At the steepest point, velocity is at its maximum (positive). At the flat point, velocity is zero (this is where distance reaches its peak). On the falling part, velocity is negative.
This section previews the two pillars of calculus. "Velocity = slope of distance" becomes the derivative in Chapter 2. "Distance = area under velocity" becomes the integral in Chapter 5. The Fundamental Theorem of Calculus (Chapter 5) formalises exactly this relationship: taking sums (integrals) is the inverse of taking differences (derivatives).
The point-slope form f(t) = f(a) + v(t - a) reappears in Section 2.3 as the tangent line approximation, and in Section 3.7 as the basis of Newton's method for solving f(x) = 0.
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