Using the Mean Value Theorem
Verify the Mean Value Theorem hypotheses before linking an average rate to an interior instantaneous rate.
College Board Topic 5.1 · CED p. 94 (PDF p. 99) · CED effective Fall 2020; project course outline 2026–27.
The Mean Value Theorem connects a whole-trip average rate to an instantaneous rate somewhere along the trip. If f is continuous on [a,b] and differentiable on (a,b), with a < b, then there is at least one c in (a,b) for which f′(c) = [f(b) − f(a)]/(b − a). The secant slope and some tangent slope must match. Check both hypotheses before using the conclusion.
Continuity includes the endpoints; differentiability is required only inside the interval. A smooth polynomial qualifies on every closed finite interval. A corner, jump, or missing point inside the interval breaks a hypothesis. When a hypothesis fails, the theorem gives no guarantee; the conclusion could still happen for a different reason. The theorem also does not promise a unique c or tell you where it is.
For f(x) = x² on [1,3], the average rate is (9 − 1)/2 = 4. Solve 2c = 4 to find c = 2. That computation illustrates the guarantee, but many applications only ask you to justify existence from endpoint data. In a driving context, a continuous differentiable position function with average velocity 50 mph has an instantaneous velocity of 50 mph somewhere in the open time interval.
A graph makes the comparison visible: the secant through the endpoints is parallel to a tangent at c. A table with only endpoints supplies the average rate, but it does not by itself establish continuity or differentiability. Use those properties when the problem states them. Rolle’s Theorem is the equal-endpoint case: the secant slope is zero.
Let's work through it
For f(x) = x³ on [−1,2], f is continuous on the closed interval and differentiable on its interior.
Why: A polynomial satisfies the two MVT hypotheses.
The secant slope is [8 − (−1)]/[2 − (−1)] = 3.
Why: The theorem compares the instantaneous rate with the endpoint average rate.
Solve f′(c) = 3c² = 3 to obtain c = ±1; only c = 1 is inside (−1,2).
Why: The theorem’s point must lie in the open interval, so the left endpoint is excluded.
At c = 1, the tangent slope equals 3.
Why: This verifies the specific point promised by the theorem.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
Which intervals are used for the two MVT hypotheses?
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These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Which intervals are used for the two MVT hypotheses?
Worked answer and reason
Continuous on [a,b], differentiable on (a,b).
Endpoint differentiability is not a hypothesis.
Retrieval 2
If endpoint values match, what derivative value is guaranteed?
Worked answer and reason
Zero somewhere in (a,b), if the hypotheses hold.
The endpoint secant slope is zero.
Retrieval 3
Does MVT guarantee exactly one matching tangent?
Worked answer and reason
No, at least one.
Existence does not imply uniqueness.
Catch the tempting mistake
- Endpoint data prove a function is smooth.
- They supply an average rate; smoothness must be stated or justified separately.
- If MVT fails to apply, no matching slope exists.
- The theorem is inconclusive when a hypothesis fails.
Make the picture move
Explore → predict → explain
A derivative is a limit of slopes.
Try first. At the selected a, predict the tangent slope. Which side’s secant lies above it, and which lies below?
Scroll sideways to see the full graph on a small screen. The table below gives the numerical values.
- Function: x²
- Selected secant
- Exact tangent
Right secant slope 2.5 at a = 1. The derivative is 2.
Compare the numbers
| Distance |h| | Left secant slope | Right secant slope |
|---|---|---|
| 0.5 | 1.5 | 2.5 |
| 0.25 | 1.75 | 2.25 |
| 0.1 | 1.9 | 2.1 |
| 0.05 | 1.95 | 2.05 |
What’s going on? For x², the difference quotient simplifies to 2a + h. As h approaches zero from either side, the secant slope approaches 2a.
Keep in mind. The controls keep h nonzero so that the quotient is defined. A small h is an approximation; the limiting calculation supplies the exact derivative.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
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+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 5 problem 1 · No calculator · analytic
- Unit 5 problem 2 · No calculator · analytic
- Unit 5 problem 3 · No calculator · numerical
For the teacher
- Use a quick two-column check: hypotheses and conclusion.
- Ask learners to distinguish an existence argument from solving for all possible c-values.
Glossary and deeper reading · 7 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Instantaneous rate of change
An instantaneous rate of change is the limiting average rate around one input when that limit exists. For a differentiable function, it is its derivative at that input. For example, position s(t)=t² meters has instantaneous rate 4 meters per second at t=2 seconds. The rate’s units are output units per input unit.
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Differentiability
Differentiability means that a function has a finite derivative at the point under discussion. Differentiability at a point implies continuity there, but continuity alone does not imply differentiability. For example, |x| is continuous at 0 but its left and right tangent slopes differ, so it is not differentiable there. Corners, cusps, vertical tangents, and discontinuities can prevent a finite derivative.
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Continuity
A function is continuous at an input when its value exists, its finite limit exists, and those two values agree. At a domain endpoint, continuity uses the relevant one-sided approach. For example, x² is continuous at 3 because both its value and its nearby limit are 9. Continuity on an interval requires the appropriate condition at every input in that interval.
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Interval
An interval is a set of real numbers containing every number between any two of its members. Parentheses exclude an endpoint, while square brackets include it. For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1. An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.
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Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.
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Velocity in calculus
In one-dimensional motion, velocity is the derivative of position with respect to time. Its sign records direction relative to the chosen positive axis. For example, s(t)=t²−4t meters gives velocity 2t−4 meters per second. Speed is the magnitude of velocity and therefore cannot be negative.
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Rolle’s theorem
Rolle’s theorem is the equal-endpoint case of the Mean Value Theorem. A function continuous on [a,b], differentiable on (a,b), and satisfying f(a)=f(b) has an interior point with derivative 0. For example, x² on [−1,1] has equal endpoint values and derivative 0 at x=0. A sharp corner can break the differentiability hypothesis even when the endpoints agree.