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Calc BC
Concepts
| Question | Answer |
|---|---|
| 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
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