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1.1 Introducing Calculus: Can Change Occur at an Instant?

Syllabus
2020
Topic
1.1
Level
—

Learning objectives

—CHA-1.A—Interpret the rate of change at an instant in terms of average rates of change over intervals containing that instant• CHA-1.A Interpret the rate of change at an instant in terms of average rates of change over intervals containing that instant.• CHA-1.A.1 Calculus uses limits to understand and model dynamic change.• CHA-1.A.2 Because an average rate of change divides the change in one variable by the change in another, the average rate of change is undefined at a point where the change in the independent variable would be zero.• CHA-1.A.3 The limit concept allows us to define instantaneous rate of change in terms of average rates of change.• Enduring understanding CHA-1: Calculus allows us to generalize knowledge about motion to diverse problems involving change.

From Average Change to Change at an Instant

An instantaneous rate of change is defined as the limit of average rates of change over intervals that contain the instant and shrink toward zero width.

\text{Average rate on }[a,b]=\frac{f(b)-f(a)}{b-a}

At a single point, a=ba=b, so the denominator b−ab-a is zero and the average-rate quotient is undefined. Calculus does not divide by zero; it uses quotients from nearby intervals whose widths are nonzero.

\text{Instantaneous rate at }x=c=\lim_{h\to0}\frac{f(c+h)-f(c)}{h}

For f(t)=t2f(t)=t^2, the average rate over [3,3+h][3,3+h] is [(3+h)2−32]/h=(6h+h2)/h=6+h[(3+h)^2-3^2]/h=(6h+h^2)/h=6+h for h≠0h\ne0. As h→0h\to0, these average rates approach 66, so the instantaneous rate at t=3t=3 is 66 output-units per input-unit.

Choose intervals containing the instant → compute each average rate → shrink the interval from either side → if the rates approach one value, that limit is the instantaneous rate.

The limit h→0h\to0 does not mean set h=0h=0 in the original quotient. It asks what the quotient approaches for nonzero hh values arbitrarily close to zero.