AP Exams
30 Cards — Free
AP Calculus AB
Key concepts for AP Calculus AB including limits, derivatives, integrals, and the fundamental theorem.
All 30 Cards
Limit Definition
The value a function approaches as x approaches a specific value; lim x→a f(x) = L
Continuity
f(x) is continuous at x=a if: f(a) exists, lim x→a f(x) exists, and lim x→a f(x) = f(a)
Derivative Definition
f'(x) = lim h→0 [f(x+h) - f(x)] / h; represents instantaneous rate of change; slope of tangent line
Power Rule
d/dx [xⁿ] = nxⁿ⁻¹; example: d/dx [x³] = 3x²
Product Rule
d/dx [f·g] = f'g + fg'; 'first times derivative of second plus second times derivative of first'
Quotient Rule
d/dx [f/g] = (f'g - fg') / g²; 'low d-high minus high d-low over low squared'
Chain Rule
d/dx [f(g(x))] = f'(g(x)) · g'(x); derivative of outer times derivative of inner
d/dx [sin x]
cos x
d/dx [cos x]
-sin x
d/dx [tan x]
sec²x
d/dx [eˣ]
eˣ (the only function that is its own derivative)
d/dx [ln x]
1/x
Implicit Differentiation
Differentiate both sides with respect to x; treat y as a function of x; multiply dy/dx when differentiating y terms
Related Rates
Find how rates of change are connected; differentiate equation with respect to time; solve for unknown rate
Mean Value Theorem
If f is continuous on [a,b] and differentiable on (a,b), there exists c where f'(c) = [f(b)-f(a)]/(b-a)
Critical Points
Where f'(x) = 0 or f'(x) is undefined; candidates for local maxima and minima
First Derivative Test
f' changes + to - at c → local max. f' changes - to + at c → local min
Second Derivative Test
f''(c) > 0 → concave up → local min. f''(c) < 0 → concave down → local max
Inflection Point
Where concavity changes; f''(x) = 0 or undefined AND concavity actually changes
Antiderivative
Reverse of derivative; F'(x) = f(x); ∫f(x)dx = F(x) + C; always add constant C
∫xⁿ dx
xⁿ⁺¹/(n+1) + C (n ≠ -1); reverse of power rule
∫1/x dx
ln|x| + C
∫eˣ dx
eˣ + C
∫sin x dx
-cos x + C
∫cos x dx
sin x + C
U-Substitution
∫f(g(x))g'(x)dx; let u = g(x), du = g'(x)dx; reverse of chain rule
Fundamental Theorem (Part 1)
If F(x) = ∫ₐˣ f(t)dt, then F'(x) = f(x); derivative of an integral returns the original function
Fundamental Theorem (Part 2)
∫ₐᵇ f(x)dx = F(b) - F(a); definite integral = antiderivative evaluated at endpoints
Area Under Curve
∫ₐᵇ f(x)dx gives signed area between f(x) and x-axis from a to b; negative below x-axis
Average Value
Average value of f on [a,b] = (1/(b-a)) · ∫ₐᵇ f(x)dx
Study this deck on the go with Stacked — the AI-powered flashcard app.
Get Stacked Free →