{
Ch. 5 Review: Integrals
AP Calculus
5.2: The Differential dy5.2: Linear Approximation5.3: Indefinite Integrals5.4: Riemann Sums (Definite Integrals)5.5: Mean Value Theorem/Rolleโs Theorem
Ch. 5 Test Topics
dx & dy: change in x and y for tangent (derivative)
The Differential dy
Tangent line
Find the differential dy:y =
dy = (6x โ 4) dx
๐๐ฆ๐๐ฅ
= ๐ โฒ (๐ฅ ) , ๐ ๐๐๐ฆ= ๐ โฒ (๐ฅ) โ๐๐ฅ
Linear Approximation
Write the equation of the line that bestfits at x = 2. Then find dx, and dy if f(2.01) is approximated.
Equation:
dx
dy
Linear Approximation
Write the equation of the line that bestfits at x = 2. Then find dx, and dy if f(2.01) is approximated.Point of tangency: f(2) = -2 Slope of tangent (deriv):
yโ = 6x โ 7 when x = 2 5
Sub into pt-slope equation:y โ
y + 2 = 5(x โ 2) y = 5x โ 12 If x = 2.01, y = -1.95
: Function change in y: f(2.01) โ f(2) = .0503dx: Tangent line change in x -- 2.01 โ 2 = .01dy: Tangent line change in y for x = 2 to 2.01: -1.95 - -2 = .05 or dy = fโ(x) dx at x = 2 (6(2) โ 7)(.01) = .05
If a function is continuous and differentiable on the interval [a, b], then there is at least one point x = c at which the slope of the tangent equals the slope of the secant connecting f(a) and f(b)
Mean Value Theorem
If a function f is:1) Differentiable for all values of x in the
open interval (a, b) and2) Continuous for all values of x in the
closed interval [a, b]
Then there is at least one number x = c in (a, b) such that
Mean Value Theorem (MVT)
fโ(c) =
If a function is differentiable and continuous on the interval [a, b], and f(a) = f(b) = 0, then there is at least one value x = c such that fโ(c) = 0.
Rolleโs Theorem
Remember โ Function must be CONTINUOUS and DIFFERENTIABLE on interval! Otherwise, conclusion of MVT may not be met.
Mean Value Theorem
Integrals Self-Quiz
โซ 8๐ฅ1 /3๐๐ฅ=ยฟยฟโซ (5๐ฅ4+1 )๐๐ฅ=ยฟยฟโซ(7 ๐ฅ+3)8๐๐ฅ=ยฟยฟ
โซ5 ๐ ๐๐2 ๐ฅ ๐๐ฅ=ยฟยฟ
โซ ๐ ๐๐ 5 ๐ฅ tan5 ๐ฅ ๐๐ฅ=ยฟยฟ
Integrals Self-Quiz
โซ 8๐ฅ1 /3๐๐ฅ=6 ๐ฅ4 /3+๐โซ (5๐ฅ4+1 )๐๐ฅ=๐ฅ5+๐ฅ+๐โซ(7 ๐ฅ+3)8๐๐ฅ=
163
(7๐ฅ+3)9+๐
โซ5 ๐ ๐๐2 ๐ฅ ๐๐ฅ=โ52cos2๐ฅ+๐
โซ ๐ ๐๐ 5 ๐ฅ tan5 ๐ฅ ๐๐ฅ=15sec 5 ๐ฅ+๐
Integrals Self-Quiz
โซ๐sin ๐ฅ๐๐๐ ๐ฅ ๐๐ฅ=ยฟยฟ
โซ๐ฅ (๐ฅ2โ3)5 ๐๐ฅ=ยฟยฟ
โซ๐๐๐ 4 ๐ฅ๐ ๐๐๐ฅ ๐๐ฅ=ยฟ ยฟ
โซ 2๐ฅ (๐ฅ3โ7 )๐๐ฅ=ยฟยฟ
Integrals Self-Quiz
โซ๐sin ๐ฅ๐๐๐ ๐ฅ ๐๐ฅ=๐๐ ๐๐๐ฅ+๐
โซ๐ฅ (๐ฅ2โ3)5 ๐๐ฅ=112
(๐ฅ2โ3)6+๐
โซ๐๐๐ 4 ๐ฅ๐ ๐๐๐ฅ ๐๐ฅ=โ15๐๐๐ 5๐ฅ+๐
โซ 2๐ฅ (๐ฅ3โ7 )๐๐ฅ=25๐ฅ5โ7 ๐ฅ2+๐
R Problems, pg. 260: R1 โR5 ab