AP® Calculus BC review sheet from Aim for Five (aimforfive.com/calc-bc/units/1/1-1)
Unit 1 · Topic 1.1
1.1 Introducing Calculus: Can Change Occur at an Instant?
Algebra can tell you how fast something changes on average between two moments. Calculus asks a harder question: how fast is it changing at one exact instant? You answer it by taking averages over shorter and shorter intervals and watching what number they close in on, which is the idea of a limit.
Key terms
- average rate of change
- instantaneous rate of change
- secant line
- tangent line
- limit
Average rate of change
The average rate of change of a function f on an interval [a, b] is the change in output divided by the change in input: (f(b) − f(a)) / (b − a). On a graph, it is the slope of the secant line, the straight line through the two points (a, f(a)) and (b, f(b)).
Its units are output units per input unit. If f(t) is distance in miles and t is time in hours, the average rate of change is in miles per hour. That is the average velocity over the trip.
The problem with one instant
A speedometer shows your speed right now, not over a stretch of time. But the average rate formula needs two different times. If you plug in a = b, you get (f(a) − f(a)) / (a − a) = 0/0, which means nothing.
Calculus gets around this. Instead of setting the interval width to 0, you make it very small, like 0.1, then 0.01, then 0.001, and look at the trend. If the average rates settle toward one number, that number is the instantaneous rate of change, the rate at that single moment.
Secant lines become a tangent line
Picture a curve and a fixed point P on it. Draw a secant line from P to a second point Q on the curve. Now slide Q along the curve toward P. The secant line pivots, and as Q gets very close to P, the secant lines line up with one special line: the tangent line at P.
The tangent line is the line that just touches the curve at P and matches its direction there. Its slope is the instantaneous rate of change at P. In Unit 2, that slope gets a name: the derivative.
| Idea | Uses | On a graph |
|---|---|---|
| Average rate of change | Two points | Slope of a secant line |
| Instantaneous rate of change | One point (as a limit) | Slope of the tangent line |
Why limits come next
The phrase “the value the average rates approach” is exactly what a limit is. The rest of Unit 1 makes that idea precise: how to write it, how to read it from graphs and tables, and how to compute it with algebra.
You won't compute instantaneous rates with formulas in this topic. The goal is the big picture: an instantaneous rate is the limit of average rates.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Average velocities closing in on an instant
A ball is thrown upward. Its height after t seconds is h(t) = 64t − 16t² feet. Find the average velocity on [1, 2], [1, 1.1], [1, 1.01] and [1, 1.001]. What does the velocity at t = 1 appear to be?
Show the solutionHide the solution
- Step 1: Average velocity is (h(b) − h(1)) / (b − 1). First, h(1) = 64 − 16 = 48.
- Step 2: On [1, 2]: h(2) = 128 − 64 = 64, so (64 − 48) / 1 = 16 ft/s.
- Step 3: On [1, 1.1]: h(1.1) = 70.4 − 19.36 = 51.04, so (51.04 − 48) / 0.1 = 30.4 ft/s.
- Step 4: On [1, 1.01]: h(1.01) = 48.3184, so 0.3184 / 0.01 = 31.84 ft/s.
- Step 5: On [1, 1.001]: h(1.001) = 48.031984, so 0.031984 / 0.001 = 31.984 ft/s.
- Step 6: The averages are climbing toward 32 and getting closer each time the interval shrinks.
Answer: The average velocities are 16, 30.4, 31.84 and 31.984 ft/s. The instantaneous velocity at t = 1 appears to be 32 ft/s.
- Example 2
Average rate from a table, with units
W(t) is the amount of water in a tank, in gallons, t minutes after a drain opens. W(4) = 104 and W(10) = 92. Find the average rate of change of W on [4, 10] and explain what it means.
Show the solutionHide the solution
- Step 1: Use output change over input change, in the same order on top and bottom: (W(10) − W(4)) / (10 − 4).
- Step 2: That is (92 − 104) / 6 = −12 / 6 = −2.
- Step 3: The units are gallons per minute. The negative sign means the amount is going down.
Answer: −2 gallons per minute: from t = 4 to t = 10 minutes, the water in the tank decreased by an average of 2 gallons per minute.
- Example 3
Trap: an average rate of 0 doesn't mean no motion
For the same ball, h(t) = 64t − 16t², find the average velocity on [0, 4]. Does the answer mean the ball didn't move?
Show the solutionHide the solution
- Step 1: h(0) = 0 and h(4) = 256 − 256 = 0.
- Step 2: Average velocity = (h(4) − h(0)) / (4 − 0) = 0/4 = 0 ft/s.
- Step 3: But the ball rose to a height of 64 feet at t = 2 and then fell back. The average rate of change only compares the two endpoints, so it can't see what happened in between.
Answer: The average velocity is 0 ft/s, but the ball was moving the whole time. An average rate of change ignores everything between the endpoints.
Common mistakes
- Subtracting in opposite orders on the top and bottom, like (W(4) − W(10)) / (10 − 4). Keep the later point first in both places, or the first point first in both.
- Calling an average rate an instantaneous rate. A slope between two points is an average, even if the points are close.
- Dropping units or the sign. A rate of −2 gallons per minute means the amount is decreasing, and the answer needs both pieces.
On the exam
- Expect to compute an average rate of change from a table or a formula and to say what it means with units. The exam asks for that interpretation again and again in later units.
- When a question says “at t = 3” it wants an instantaneous rate. When it says “over the interval” or “from t = 1 to t = 3” it wants an average.
Connected topics
Videos
Check yourself
4 questions on 1.1 Introducing Calculus: Can Change Occur at an Instant?. Pick an answer to see if you got it, and why.
For f(x) = x³, the average rate of change of f over the interval [1, 1 + h] is ((1 + h)³ − 1)/h for h ≠ 0. As h approaches 0, what value do these average rates of change approach?
The temperature, in degrees Fahrenheit, of a room is modeled by T(t) = 60 + 15 sin(t/2), where t is measured in hours. The average rate of change of T over [3, 3 + h] is computed for h = 0.1, 0.01 and 0.001. These values approach which of the following, in degrees Fahrenheit per hour?
| h | Average velocity on [3, 3 + h] (m/s) |
|---|---|
| −0.1 | 28.91 |
| −0.01 | 29.351 |
| −0.001 | 29.3951 |
| 0.001 | 29.4049 |
| 0.01 | 29.449 |
| 0.1 | 29.89 |
Invented data: average velocity of a falling ball over [3, 3 + h]
A ball is dropped from a tall tower, and the distance it has fallen after t seconds is s(t) = 4.9t² meters. The table gives the average velocity of the ball, in meters per second, over the time interval between t = 3 and t = 3 + h for several values of h. Based on the table, which of the following is the best estimate of the instantaneous velocity of the ball at t = 3 seconds?
Which of the following is the best interpretation of the value found in the previous question?
0 of 4 answered