Stewart Calculus · 8th Edition
Chapter 4: Applications of Differentiation
Every key formula from Stewart Calculus Chapter 4, in one searchable page. Click a card to study it — worked examples included.
4.1 · MAXIMUM & MINIMUM VALUES
Extreme Value Theorem
If f is continuous on [a, b], it attains an absolute maximum M and an absolute minimum m somewhere on [a, b].
Fermat’s Theorem (c critical if f′(c) = 0 or DNE)
Local max/min at interior c ⇒ f′(c) = 0
Closed Interval Method
Evaluate f at every critical number in (a, b) and at a, b — largest value = abs max, smallest = abs min.
WORKED EXAMPLE
f(x) = x³ − 3x² + 1 on [−½, 4]: f′ = 3x(x − 2) = 0 ⇒ x = 0, 2.
f(−½) = ⅛, f(0) = 1, f(2) = −3, f(4) = 17 ⇒ abs max 17 at x = 4, abs min −3 at x = 2.
f(−½) = ⅛, f(0) = 1, f(2) = −3, f(4) = 17 ⇒ abs max 17 at x = 4, abs min −3 at x = 2.
KEY NOTES
- Extrema often sit at an endpoint — never skip f(a) and f(b).
4.2 · ROLLE’S & MEAN VALUE
Rolle’s Theorem
f continuous on [a,b], differentiable on (a,b), f(a) = f(b) ⇒ ∃c ∈ (a,b) with f′(c) = 0.
Mean Value Theorem
∃c ∈ (a,b) with f′(c) = f(b) − f(a)b − a
MVT is “tilted Rolle”: avg rate = instant. rate.
KEY NOTES
- Both hypotheses are essential: continuity on closed [a,b] + differentiability on open (a,b).
4.3 · SHAPE OF A GRAPH
Increasing / Decreasing & First Derivative Test
f′ > 0 ⇒ increasing · f′ < 0 ⇒ decreasing
At critical c: f′ changes +→− ⇒ local max; −→+ ⇒ local min; no change ⇒ neither.
Concavity & Second Derivative Test
f′′ > 0 ⇒ concave up ∪ · f′′ < 0 ⇒ concave down ∩
Inflection point: concavity changes (f′′ = 0 or DNE). If f′(c) = 0: f′′(c) > 0 ⇒ local min, f′′(c) < 0 ⇒ local max.
KEY NOTES
- f′′ = 0 alone ≠ inflection — concavity must actually change sides.
4.4 · L’HÔPITAL’S RULE
The rule
00 or ∞∞ ⇒ lim fg = lim f′g′
Indeterminate forms
00, ∞∞, 0 · ∞, ∞ − ∞, 0⁰, 1∞, ∞⁰ — rewrite as a quotient first (take ln for powers).
WORKED EXAMPLE
limx→0 ex − 1 − xx² = 00 ⇒ lim ex − 12x ⇒ lim ex2 = ½.
KEY NOTES
- Confirm 00 or ∞∞ FIRST — otherwise the rule is invalid.
4.5–4.7 · CURVE SKETCHING& OPTIMIZATION
Curve-sketching checklist
- Domain, intercepts, symmetry, asymptotes
- Inc/dec + max/min via f′; concavity + inflection via f′′
Optimization steps
- Draw & label; objective Q + constraint; eliminate a variable
- Optimize on its domain: critical numbers (+ endpoints if closed)
WORKED EXAMPLE
Fence 100 ft around a rectangle: y = 50 − x, A = x(50 − x).
A′ = 50 − 2x = 0 ⇒ x = 25 ft, y = 25 ft ⇒ max area 625 ft² (a square!).
A′ = 50 − 2x = 0 ⇒ x = 25 ft, y = 25 ft ⇒ max area 625 ft² (a square!).
4.9 · ANTIDERIVATIVES
Definition
F′(x) = f(x) ⇒ F is an antiderivative of f
General form: F(x) + C — always add the constant!
Common antiderivatives
xn → xn+1n+1 (n ≠ −1) · 1x → ln|x| · ex → ex
sin x → −cos x · cos x → sin x
sin x → −cos x · cos x → sin x
WORKED EXAMPLE
f(x) = x² + 3x ⇒ F(x) = x³3 + 3x²2 + C.
Check: F′ = x² + 3x ✓.
Check: F′ = x² + 3x ✓.
Any two antiderivatives differ by a constant: G′ = F′ ⇒ G = F + C.
Geometrically: their graphs are vertical shifts of each other.
Geometrically: their graphs are vertical shifts of each other.
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