ap_calc_bc_sss_1_-_the_big_theorems_evt_ivt_mvt_ftc

AP Calculus BC Saturday Study Session #1: The “Big” Theorems

Overview of Theorems

  • Students should be able to apply key theorems (EVT, IVT, MVT, FTC) on the AP Calculus Exam;

  • Proofs of the theorems are not required.

Extreme Value Theorem (EVT)

  • Formal Statement: If a function is continuous on a closed interval [a, b], then:

    1. There exists at least one number c in [a, b] such that f(c) is the maximum value on [a, b].

    2. There exists at least one number d in [a, b] such that f(d) is the minimum value on [a, b].

  • Translation: A continuous function on a closed interval achieves both maximum and minimum values on that interval.

  • Important Notes:

    • Maximum and minimum values may occur multiple times;

    • Extreme values often occur at the endpoints of the interval;

    • For constant functions, maximum and minimum values are identical.

Intermediate Value Theorem (IVT)

  • Formal Statement: If a function f is continuous on a closed interval [a, b] and f(a) does not equal f(b), then for every value k between f(a) and f(b), there exists at least one value c in (a, b) with f(c) = k.

  • Translation: A continuous function will take on every value between any two values it achieves.

Mean Value Theorem (MVT)

  • Formal Statement: If a function f is continuous on [a, b] and differentiable on (a, b), then there exists at least one number c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

  • Translation: In a continuous and differentiable function, there is at least one point where the instantaneous rate of change equals the average rate of change.

Rolle’s Theorem

  • A special case of MVT where f(a) = f(b).

  • Formal Statement: If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists at least one number c in (a, b) such that f'(c) = 0.

  • Translation: For functions equal at two endpoints, there exists at least one horizontal tangent in-between.

Fundamental Theorem of Calculus (FTC)

  • Assumptions: Let f, g, and h be differentiable functions and h(x) be an antiderivative of f(x).

  • First Fundamental Theorem of Calculus (1st FTC):

    • ∫_a^b f(x) dx = F(b) - F(a)

    • F is an antiderivative of f.

  • Second Fundamental Theorem of Calculus (2nd FTC):

    • If F is an antiderivative of f, then ∫_a^b f(x) dx = F(b) - F(a).

    • Also includes a variation of the chain rule.

Multiple Choice Questions on Theorems

EVT & IVT Questions

  1. 1997 #81 (BC): Given continuous f on [-3,6] where f(-3) = -1 and f(6) = 3, applying IVT guarantees:

    • a. f(0) = 0

    • b. f'(c) = 4 for at least one c between -3 and 6.

    • c. -1 ≤ f(x) < 3 for all x in [-3,6].

    • d. f(c) = 1 for at least one c in [-3,6].

    • e. f(c) = 0 for at least one c in [-1,3].

  2. 1998 #91 (AB): Given differentiable f on (1,10) where f(2) = -5, f(5) = 5, f(9) = -5, the must-be-true statements include:

    • I. f has at least 2 zeros.

    • II. Graph of f has at least one horizontal tangent.

    • III. For some c in (2,5), f(c) = 3.

MVT Questions

  1. 2003 #83 (BC): Given table of a function f over [0,4], true statements include:

    • a. Minimum value in [0,4] is 2.

    • e. There exists c, with 0 < c < 4, where f'(c) = 0.

  2. 1998 #4 (AB): For a continuous f on [a, b] and differentiable f on (a, b), which could be false?

    • b. f'(c) = 0 for some c in (a,b).

Free Response Questions

2007 #3 (AB):

  • Given differentiable functions f and g with provided values at specific points, discuss existence of r where h(r) = −5 and c where h'(c) = −5.

2011B #5 (BC):

  • Discuss velocity of Ben on a unicycle based on the rates provided.

Example of Application of IVT:

  • For an interval where rates of two different processes are compared, justify equal rates at a particular time using IVT.

Additional Practice Problems

  • See multiple choice from previous AP tests for topics covering EVT, IVT, MVT, and FTC.