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# Calc BC

### Concepts

When calculus was developed 17th century
4 major problems 17th century mathematicians worked on Tangent line, Velocity, Max/min (optimization), Area
Slope A rate of change if independent axis has a different unit than dependent axis
Average rate of change (geometrically speaking) Slope of the secant line
Instantaneous rate of change (geometrically speaking) Slope of the tangent line
Instantaneous rate of change (calc concept) 1st derivative of a function
Difference quotient (f(x+h)-f(x))/h
Instantaneous rate of change (formula) limit as h->0 of the difference quotient
First derivative notation f'(x)=y'=(dy)/(dx)=d/(dx) (f)
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.
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.
How the total "area" can be found, using the definite integral Increase n
Trapezoidal rule, using the definite integral Δx[f(x1)/2+f(x2)+f(x3)+...+f(xn)/2]
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
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
Newton and Leibniz Inventors of calculus
Cauchy Formally introduced limits
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.
Three reasons limits fail to exist Unbounded behavior; graph: step discontinuity (limits from either side are not equal); oscillating
The limit of a sum The sum of the limits
The limit of a difference The difference of the limits
The limit of a constant times a function The constant times the limit of a function
The limit of a product The product of the limits
The limit of a quotient The quotient of the limits (provided that the denominator ≠ 0)
The limit of a function raised to a power, n The limit raised to the power,n
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).
Function f is continuous on (a,b) if... f is continuous at all points in (a,b) (know the behavior of the function).
Function f is continuous on [a,b] if... f is continuous on (a,b) and f is continuous at x=a and x=b.
When the limit of a function = 0/c (where c≠0) The limit of the function = 0
When the limit of a function = c/0 (where c≠0) The limit exhibits unbounded behavior and does not exist
When the limit of a function = 0/0 It is indeterminate (should find another method to evaluate limit)
When the limit of a function = c/∞ It is essentially 0.
When the limit of a function = ∞/∞ It is indeterminate.
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.
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.)
Formal definition of the derivative at point x=c f'(c)=(f(x)-f(c))/(x-c)
Formal definition of the derivative function for f(x)=y limit as h->0 of the difference quotient
Local (relative) extrema The graph is changing from increasing to decreasing (or vice versa). f'(a)=0
Absolute (global) extrema f(b) is the maximum (or minimum) output for f.
Concave up The tangent lines to the graph lie BELOW the graph.
Concave down The tangent lines to the graph lie ABOVE the graph.
Inflection points The graph is changing concavity and the tangent line crosses the graph.
Critical point f'(c)=0 or f(c) is undefined, provided c is in the domain of f.
The limit as x->0 of sinx/x 1
The limit as x->0 of (1-cosx)/x 0
Power Rule for f(x)=x^n nx^(n-1)
Antiderivative of a polynomial function f(x)=x^k 1/(k+1) x^(k+1) +c
Chain rule for g(h(x)) g'(h(x))*h'(x)
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.
Implicitly differentiating Differentiating a function where the output is part of a composite funciton
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).
The first derivative of b^x b^x * lnb
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.
Created by: pogs89