Concepts
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| When calculus was developed | 17th century
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| 4 major problems 17th century mathematicians worked on | Tangent line, Velocity, Max/min (optimization), Area
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| Slope | A rate of change if independent axis has a different unit than dependent axis
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| Average rate of change (geometrically speaking) | Slope of the secant line
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| Instantaneous rate of change (geometrically speaking) | Slope of the tangent line
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| Instantaneous rate of change (calc concept) | 1st derivative of a function
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| Difference quotient | (f(x+h)-f(x))/h
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| Instantaneous rate of change (formula) | limit as h->0 of the difference quotient
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| First derivative notation | f'(x)=y'=(dy)/(dx)=d/(dx) (f)
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| Definition of the derivative | The instantaneous rate of change of the dependent variable with respect to the independent variable as the change in the independent variable approaches 0.
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| Definite integral | A way to find the product of (b-a)/n, where n is the number of rectangles, hence (b-a)/n=Δx, and f(x), even if f(x) is not constant.
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| How the total "area" can be found, using the definite integral | Increase n
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| Trapezoidal rule, using the definite integral | Δx[f(x1)/2+f(x2)+f(x3)+...+f(xn)/2]
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| If there exists a removable discontinuity in the graph of f(x)=y, at x=c... | The limit as x->c of f(x) exists
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| If there exists a nonremovable discontinuity in the graph of f(x)=y, at x=c... | The limit as x->c of f(x) does not exist
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| Newton and Leibniz | Inventors of calculus
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| Cauchy | Formally introduced limits
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| Formal definition of a limit | L is the limit of f(x) as x approaches c if and only if for any positive number ε, no matter how small, there is a number δ such that if x is within δ units of c, but not equal to c, then f(x) is within ε units of L.
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| Three reasons limits fail to exist | Unbounded behavior; graph: step discontinuity (limits from either side are not equal); oscillating
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| The limit of a sum | The sum of the limits
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| The limit of a difference | The difference of the limits
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| The limit of a constant times a function | The constant times the limit of a function
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| The limit of a product | The product of the limits
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| The limit of a quotient | The quotient of the limits (provided that the denominator ≠ 0)
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| The limit of a function raised to a power, n | The limit raised to the power,n
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| If a function is continuous at a point, x=c, then... | f(c) is defined; the limit as x->c of f(x) exists; the limit as x->c of f(x)=f(c).
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| Function f is continuous on (a,b) if... | f is continuous at all points in (a,b) (know the behavior of the function).
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| Function f is continuous on [a,b] if... | f is continuous on (a,b) and f is continuous at x=a and x=b.
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| When the limit of a function = 0/c (where c≠0) | The limit of the function = 0
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| When the limit of a function = c/0 (where c≠0) | The limit exhibits unbounded behavior and does not exist
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| When the limit of a function = 0/0 | It is indeterminate (should find another method to evaluate limit)
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| When the limit of a function = c/∞ | It is essentially 0.
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| When the limit of a function = ∞/∞ | It is indeterminate.
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| The Intermediate Value Theorem (IVT) | If f is continuous for all x in [a,b] and y is a number between f(a) and f(b), then there exists a number x=c in (a,b) for which f(c)=y.
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| Extreme Value Theorem | If f is continuous on[a,b], then f assumes both a maximum and a minimum value provided f is not constant on [a,b]. (They might occur @ x=a or x=b.)
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| Formal definition of the derivative at point x=c | f'(c)=(f(x)-f(c))/(x-c)
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| Formal definition of the derivative function for f(x)=y | limit as h->0 of the difference quotient
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| Local (relative) extrema | The graph is changing from increasing to decreasing (or vice versa). f'(a)=0
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| Absolute (global) extrema | f(b) is the maximum (or minimum) output for f.
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| Concave up | The tangent lines to the graph lie BELOW the graph.
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| Concave down | The tangent lines to the graph lie ABOVE the graph.
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| Inflection points | The graph is changing concavity and the tangent line crosses the graph.
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| Critical point | f'(c)=0 or f(c) is undefined, provided c is in the domain of f.
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| The limit as x->0 of sinx/x | 1
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| The limit as x->0 of (1-cosx)/x | 0
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| Power Rule for f(x)=x^n | nx^(n-1)
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| Antiderivative of a polynomial function f(x)=x^k | 1/(k+1) x^(k+1) +c
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| Chain rule for g(h(x)) | g'(h(x))*h'(x)
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| The Squeeze Theorem | If h(x)≤f(x)≤g(x) for all x in an open interval containing c, x≠c, and if the limit as x->c of h(x) = the limit as x->c of g(x) = L, then the limit as x->c of f(x)=L.
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| Implicitly differentiating | Differentiating a function where the output is part of a composite funciton
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| If f(x)=g(h(x)) and g and f are continuous and if the limit as x->c of h(x)=L, then... | The limit as x->c of f(x)=g(the limit as x->c of h(x))=g(L).
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| The first derivative of b^x | b^x * lnb
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| The product rule | The derivative of a product is the derivative of the first function * the second function + the derivative of the 2nd function * the first function.
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