Math: Calculus 1 Fundamentals
Limits, derivatives, and basic integrals — the foundation of calculus.
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What is the value of the limit as x approaches c of a constant function f(x) = k?
The limit of a constant function f(x) = k as x approaches any value c is simply the constant k.
What is the Power Rule for finding the derivative of f(x) = x^n?
The Power Rule states that the derivative of f(x) = x^n is f'(x) = n*x^(n-1).
What three conditions must be met for a function f(x) to be continuous at a point x = a?
For a function to be continuous at x = a: 1) f(a) must be defined, 2) the limit of f(x) as x approaches a must exist, and 3) the limit must equal f(a).
What is the derivative of the trigonometric function f(x) = cos(x)?
The derivative of f(x) = cos(x) is f'(x) = -sin(x).
What is the derivative of the natural exponential function f(x) = e^x?
The derivative of f(x) = e^x is simply f'(x) = e^x.
What is the general antiderivative of f(x) = x^n, where n is not equal to -1?
The general antiderivative of x^n is (x^(n+1))/(n+1) + C, where C is the constant of integration.
What is the formal limit definition of the derivative of a function f(x)?
The formal limit definition of the derivative is f'(x) = lim (h -> 0) [f(x+h) - f(x)] / h.
What is the Product Rule for finding the derivative of the product of two functions, u(x) and v(x)?
The Product Rule states that the derivative of u(x)v(x) is u'(x)v(x) + u(x)v'(x).
What is the Quotient Rule for finding the derivative of f(x) = u(x) / v(x)?
The Quotient Rule is f'(x) = [u'(x)v(x) - u(x)v'(x)] / [v(x)]^2.
How do you apply the Chain Rule to find the derivative of a composite function f(g(x))?
The Chain Rule states that the derivative of f(g(x)) is f'(g(x)) * g'(x).
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