Limits and the definition of the derivative
PHYS 110 · Differential Calculus
A derivative measures instantaneous rate of change. This lesson builds the derivative from average rates, limits, secant lines, and tangent lines.
Key equations
v_{avg} = \frac{\Delta x}{\Delta t}\frac{dx}{dt} = \lim_{\Delta t \to 0}\frac{x(t+\Delta t)-x(t)}{\Delta t}v(t) = \frac{dx}{dt}a(t) = \frac{dv}{dt}a(t) = \frac{d^2x}{dt^2}Learning objectives
- Explain average and instantaneous rates of change.
- Interpret the derivative as a limit.
- Connect derivatives to tangent-line slopes.
- Relate position, velocity, and acceleration using derivatives.
Average rate of change
Calculus begins with the idea of change. If position changes from to during a time interval from to , the average velocity is
v_{avg} = rac{Delta x}{Delta t} = rac{x_2-x_1}{t_2-t_1}
This is an average rate of change. It tells how much position changed per unit time over an interval.
Graphically, average rate of change is the slope of a secant line, a line connecting two points on a curve.
Instantaneous rate of change
Physics often needs the rate of change at a single instant. A car's speedometer shows something closer to instantaneous speed than average speed over a long trip. To find an instantaneous rate, we shrink the time interval.
The derivative is defined as a limit:
rac{dx}{dt} = lim_{Delta t o 0} rac{x(t+Delta t)-x(t)}{Delta t}
This expression says: calculate the average rate of change over a small interval, then examine what value it approaches as the interval becomes extremely small.
What a limit means
A limit describes the value a function approaches as the input approaches some value. It does not always require the function to actually equal that value at the point. Limits allow calculus to handle instantaneous change by reasoning about nearby values.
In the derivative definition, approaches zero, but we do not simply set in the fraction because that would produce division by zero. Instead, we simplify or reason about the expression and find the value it approaches.
Tangent lines
As the two points on a secant line move closer together, the secant line approaches a tangent line. The tangent line touches the curve locally and has the same instantaneous slope as the curve at that point.
The derivative is the slope of this tangent line. If the derivative is positive, the function is increasing. If it is negative, the function is decreasing. If it is zero, the graph is locally flat.
Position, velocity, and acceleration
In physics, if position is , velocity is the derivative of position:
v(t) = rac{dx}{dt}
Acceleration is the derivative of velocity:
a(t) = rac{dv}{dt}
Acceleration is also the second derivative of position:
a(t) = rac{d^2x}{dt^2}
This notation means position has been differentiated twice with respect to time.
Example: constant velocity motion
If
then the slope is always 3. The derivative is
rac{dx}{dt} = 3
This means velocity is constant. The intercept 2 is initial position, and the coefficient 3 is velocity.
Example: changing velocity
If
then the graph curves. The derivative is
rac{dx}{dt} = 2t
Velocity changes with time. At , velocity is 2. At , velocity is 6. The derivative captures the changing slope.
The big idea
A derivative is an instantaneous rate of change and the slope of a tangent line. Limits make this idea precise. In physics, derivatives turn position into velocity, velocity into acceleration, and changing quantities into measurable rates. Differential calculus is the mathematics of motion, variation, and local behavior.
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