Calculus BC - Formula Sheet Flashcards

Questions and Answers

What is the formal version of the definition of a derivative?

Formal definition of derivative is lim(h->0) [f(c+h) - f(c)]/h.

What is the alternate version of the definition of derivative?

Alternate definition of derivative is f'(x) = lim(h->0) [f(x+h) - f(x)]/h.

What is the definition of continuity at c?

f is continuous at c iff: i. f(c) is defined; ii. lim x->c f(x) exists; iii. lim x->c f(x) = f(c).

What is the formula for average rate of change of f(x) on [a,b]?

Average rate of change = (f(b) - f(a)) / (b - a). Signup and view all the answers

What does the Intermediate Value Theorem (IVT) state?

If f is continuous on [a,b] and k is between f(a) and f(b), then there exists c in (a,b) such that f(c) = k. Signup and view all the answers

What is the derivative of the product u and v?

d/dx [uv] = u dv/dx + v du/dx. Signup and view all the answers

What is the derivative of the quotient u and v?

d/dx [u/v] = (v du/dx - u dv/dx) / v^2. Signup and view all the answers

What is the derivative of the composition f(g(x))?

d/dx [f(g(x))] = f'(g(x)) * g'(x). Signup and view all the answers

What is the derivative of sin(u)?

d/dx [sin u] = cos u du/dx. Signup and view all the answers

What is the derivative of cos(u)?

d/dx [cos u] = -sin u du/dx. Signup and view all the answers

State Rolle's Theorem.

If f is continuous on [a,b] and differentiable on (a,b) with f(a) = f(b), then there exists c in (a,b) such that f'(c) = 0. Signup and view all the answers

State the Mean Value Theorem.

In a continuous function on a closed interval [a,b], there exists at least one c in (a,b) such that f'(c) = (f(b) - f(a)) / (b - a). Signup and view all the answers

What is a critical number?

A critical number c is where f'(c) = 0 or f' is undefined at c. Signup and view all the answers

What does the First Derivative Test state?

For a critical number c of f, if f'(x) changes from negative to positive at c, then f(c) is a relative minimum. Signup and view all the answers

What does the Second Derivative Test indicate?

If f''(c) > 0, then f(c) is a relative minimum; if f''(c) < 0, then f(c) is a relative maximum. Signup and view all the answers

What defines an inflection point?

A function has an inflection point at (c, f(c)) if f''(c) = 0 or f'' has a sign change at x=c. Signup and view all the answers

What is the integral of cos(u)?

∫ cos u du = sin u + C. Signup and view all the answers

What is the integral of sin(u)?

∫ sin u du = -cos u + C. Signup and view all the answers

What does the First Fundamental Theorem of Calculus state?

∫ from a to b f'(x)dx = F(b) - F(a). Signup and view all the answers

What does the Second Fundamental Theorem of Calculus state?

d/dx ∫ from a to x f(t)dt = f(x). Signup and view all the answers

What is the average value of f(x) on [a,b]?

Average value = (1/(b-a)) ∫ from a to b f(x)dx. Signup and view all the answers

What is the derivative of natural logarithm u?

d/dx [ln u] = 1/u du/dx. Signup and view all the answers

What is the integral of 1/u?

∫ 1/u du = ln |u| + C. Signup and view all the answers

What is the formula for displacement from t = a to t = b?

Displacement = ∫ from a to b v(t)dt. Signup and view all the answers

What is the formula for total distance traveled from t = a to t = b?

Total distance = ∫ from a to b |v(t)| dt. Signup and view all the answers

What is the Pythagorean identity for sin²(x)?

sin²(x) = 1 - cos²(x). Signup and view all the answers

What is the formula for the length of an arc for functions?

s = ∫ from a to b √(1 + [f'(x)]²) dx. Signup and view all the answers

What is the formula for speed (or magnitude of velocity vector)?

||v(t)|| = √((dx/dt)² + (dy/dt)²). Signup and view all the answers

How do you define logistic growth?

dP/dt = kP(L - P). Signup and view all the answers

What does the nth term test for divergence state?

If lim (n->∞) a(n) ≠ 0, then the series diverges. Signup and view all the answers

What are the conditions for convergence of a geometric series?

Convergence occurs if |r| < 1. Signup and view all the answers

State the Ratio Test.

Convergence: lim (n->∞) |a_n+1 / a_n| < 1; Divergence: lim (n->∞) |a_n+1 / a_n| > 1. Signup and view all the answers

What is the Taylor series for e^x?

Taylor series of e^x = 1 + x + x²/2! + x³/3! + ... for all x. Signup and view all the answers

What is the Taylor series for sin(x)?

Taylor series of sin(x) = x - x³/3! + x⁵/5! - ... for all x. Signup and view all the answers

Study Notes

Derivatives and Continuity

The formal definition of the derivative is a limit that captures the instantaneous rate of change of a function.
An alternate version of the derivative is linked to the secant line's slope approaching the tangent line's slope.
A function f is continuous at point c if:
- f(c) is defined,
- the limit of f(x) as x approaches c exists, and
- the limit equals f(c).

Theorems and Techniques

The average rate of change of f(x) on interval [a, b] is calculated by (f(b) - f(a))/(b - a).
The Intermediate Value Theorem states if f is continuous on [a, b] and k is in the interval [f(a), f(b)], there exists at least one c such that f(c) = k.
Rolle's Theorem confirms there is at least one c in (a, b) where f'(c) = 0, given f is continuous on [a, b] and differentiable on (a, b) with f(a) = f(b).
The Mean Value Theorem asserts that there exists at least one c in (a, b) where f’(c) equals the average rate of change over that interval.

Differentiation Rules

Product Rule: d/dx [uv] = u(dv/dx) + v(du/dx).
Quotient Rule: d/dx [u/v] = (v(du/dx) - u(dv/dx)) / v².
Chain Rule: d/dx [f(g(x))] = f'(g(x)) * g'(x).
Derivatives of trig functions:
- d/dx [sin(u)] = cos(u) * (du/dx)
- d/dx [cos(u)] = -sin(u) * (du/dx)
- d/dx [tan(u)] = sec²(u) * (du/dx)
- d/dx [sec(u)] = sec(u)tan(u) * (du/dx)
- d/dx [cot(u)] = -csc²(u) * (du/dx)
- d/dx [csc(u)] = -csc(u)cot(u) * (du/dx).

Critical Points and Extrema

A critical number c occurs if f is defined at c and either f'(c) = 0 or f' is undefined.
The First Derivative Test classifies critical points based on changes in sign:
- From negative to positive indicates a relative minimum.
- From positive to negative indicates a relative maximum.
The Second Derivative Test confirms:
- f''(c) > 0 implies a relative minimum.
- f''(c) < 0 implies a relative maximum.
- f''(c) = 0 is inconclusive.

Integrations and Theorems

The Grund's theorem states that the integral of f'(x) over [a, b] equals F(b) - F(a).
The second fundamental theorem of calculus states that d/dx ∫(from a to x) f(t)dt = f(x).
The average value of a function f(x) on interval [a,b] is given by (1/(b-a))∫(from a to b) f(x)dx.

Integral Results

Fundamental integrals:
- ∫cos(u) du = sin(u) + C
- ∫sin(u) du = -cos(u) + C
- ∫sec²(u) du = tan(u) + C
- ∫csc²(u) du = -cot(u) + C
- ∫e^u du = e^u + C.

Series and Convergence Tests

The nth term test for divergence states if lim (n→∞) a(n) ≠ 0, then the series diverges.
A geometric series converges if |r| < 1, diverges if |r| ≥ 1.
The ratio test helps determine convergence: lim (n→∞) |a(n+1)/a(n)| < 1 indicates convergence and > 1 indicates divergence.

Taylor Series

Taylor series expansion for e^x: ∑ (x^n/n!) for all x.
Taylor series for sin(x): ∑ (-1)^(n)(x^(2n+1)/(2n+1)!) for all x.
Taylor series for cos(x): ∑ (-1)^(n)(x^(2n)/(2n)!) for all x.
The series for ln(1+x) involves alternating terms based on x^(n)/n with the radius of convergence for -1 < x ≤ 1.

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Test your knowledge of key calculus concepts with these flashcards. This quiz covers fundamental definitions, theorems, and concepts crucial for understanding Calculus BC. Perfect for reviewing essential formulas and their applications.