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]?
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What does the Intermediate Value Theorem (IVT) state?
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What is the derivative of the product u and v?
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What is the derivative of the quotient u and v?
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What is the derivative of the composition f(g(x))?
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What is the derivative of sin(u)?
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What is the derivative of cos(u)?
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State Rolle's Theorem.
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State the Mean Value Theorem.
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What is a critical number?
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What does the First Derivative Test state?
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What does the Second Derivative Test indicate?
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What defines an inflection point?
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What is the integral of cos(u)?
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What is the integral of sin(u)?
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What does the First Fundamental Theorem of Calculus state?
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What does the Second Fundamental Theorem of Calculus state?
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What is the average value of f(x) on [a,b]?
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What is the derivative of natural logarithm u?
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What is the integral of 1/u?
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What is the formula for displacement from t = a to t = b?
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What is the formula for total distance traveled from t = a to t = b?
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What is the Pythagorean identity for sin²(x)?
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What is the formula for the length of an arc for functions?
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What is the formula for speed (or magnitude of velocity vector)?
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How do you define logistic growth?
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What does the nth term test for divergence state?
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What are the conditions for convergence of a geometric series?
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State the Ratio Test.
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What is the Taylor series for e^x?
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What is the Taylor series for sin(x)?
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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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