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Calc Exam 1
Topis 1.1 to 2.4
| Term | Definition |
|---|---|
| function | a rule that takes certain numbers as inputs & assigns to each a definition output number |
| domain | set of all input numbers |
| range | set of all output numbers |
| The Rule of Four | mathematical ideas can be represented and understood graphically, symbolically, numerically, & verbally |
| a function f is increasing if | the values of f(x) increase as x increases |
| a function f is decreasing if | the values of f(x) decrease as x increases |
| linear function | has a constant rate of change or slope |
| slope intercept form | y=mx+b |
| point-slope form | y=m(x-x0)+y0 |
| vertical intercept | where the function crosses the y-axis |
| vertical line test | if a vertical line only touches the graph at 1 point it is a function |
| exponential funciton | a function of the form P(t)=P0a^t |
| P0 | initial quantity |
| a | growth factor |
| a function has exponential growth if | a>1 |
| growth rate = | growth factor-1 |
| a function has exponential decay if | 0<a<1 |
| a function is concave up if | if its graph curves upwards |
| a function is concave down if | its graph curves downwards |
| a linear function is neither | concave up or down |
| half-life of an exponentially decaying quantity is | the time it takes the amount to be reduced by 1/2 |
| doubling time of an exponentially growing quantity is | the time it takes the amount to double |
| exponential functions are about | constant ratios |
| an exponential function can be written with base e | P(t)=P0e^kt |
| e^k | growth factor |
| k= | continuous growth/decay rate |
| when 0<e^k<1 there is | exponential decay which happens when k<0 |
| when e^k>1 there is | exponential growth which happens when k>0 |
| f(x)+/-h | shift up/down by h |
| f(x+/-h) | shift left/right by h |
| hf(x) | vertical stretch by h |
| 1/h*f(x) | vertical compression by h |
| -f(x) | vertical reflection over x-axis |
| f(-x) | horizontal reflection over y-axis |
| f(hx) | horizontal compression by factor of h or 1/h |
| a function is even if | f(x)=-f(x) |
| a function is odd if | -f(x)=f(-x) |
| a graph is symmetric about the origin if | it has only odd powers of x |
| a function is invertible if | every output value corresponds to a unique input value |
| log(x) is the inverse function of | 10^x |
| log is the power of 10 needed to get | x |
| log(x) is undefined if | x<0 or x=0 |
| ln(x) is the inverse of | e^x |
| log(A*B)= | log(A)+log(B) |
| log(A/B)= | log(A)-log(B) |
| log(A^B)= | B*log(A) |
| 10^log(x)= | x |
| log(1)= | 0 |
| log(10^x)= | x |
| e^ln(x)= | x |
| unit circle | circle with radius 1 centered at the origin |
| amplitude= | (max-min)/2 |
| period | the time it takes for the function to complete 1 cycle |
| formula for sinusoidal functions | y=amp*cos/sin(Bx)+mid |
| arcsin(y) | the x-value between -pi/2 and pi/2 that makes sin(x)=y |
| arccos(y) | the x-value between 0 & pi that makes cos(x)=y |
| Fundamental Theorem of Trig | sin^2theta+cos^2theta=1 |
| a power function has the form | f(x)=kx^p where k &p are constants |
| when considering functions of the form x^p for p, a positive integer, the shape of the graph is determined by | whether p is even or odd |
| when p is odd x^p is | an increasing odd function |
| when p is even x^p is | an decreasing even function |
| dominance | we say that a function g(x) dominates f(x) if g(x) grows much more quickly than f(x) |
| increasing exponential functions dominate | any power function |
| polynomial | the sum of power functions with non-negative integer exponents |
| for x large a polynomial looks like | its highest degree term |
| if the coefficient of a polynomials highest degree term is pos then | the polynomial approaches infinity as x grows large |
| we can use the x-intercepts or roots of a polynomial to approximate | its formula from a graph |
| rational funciton | is of the form f(x)/g(x) where f(x) and g(x) are polynomials |
| vertical asymptotes & holes | y approaches pos/neg infinity if x=c is still a root of the denominator of a function after eliminating like terms then its an asymptote otherwise it is a hole |
| horizontal asymptote | if the numerator has the highest degree the rational function goes to pos/neg infinity & has no H. asymptotes if the denominator has the highest degree y=0 if they have the same degree the asymptote is found by taking the ratio of the coefficients |
| the degree of a polynomial is | the highest power of x that appears |
| fundamental theorem of algebra | a polynomial of degree n has exactly n roots |
| root | where f(x)=0 |
| a polynomial of degree n has at most | n-1 turns where n is all powers added together |
| function has an open circle | function does not take on the value |
| function has a closed circle | function does take on the value |
| when a rational function is in indeterminate form to find the limit | you need to factor the fractions as much as possible & then cancel |
| the only foolproof method of proving a limit is | the symbolic method |
| the function f is continuous at a if | f is defined at a lim f(x) exists and x approaches a lim as x approaches a of f(x)=f(a) |
| intermediate value theorem | say f is a continuous function on the interval [a,b]. if k is between f(a) & f(b) then there is at least one number c between a & b such that f(c)=k |
| limits don't exist when | there are different left & right limits vertical asymptotes (b.c. they are undefined at that point |
| a limit is | let f be a function of x we say lim as x approaches c of f(x)=L if f(x) is "very close" to L whenever x is "close" to c |
| right hand limit | the limit of a function as x approaches a number from the right |
| left hand limit | the limit of a function as x approaches a number from the left |
| if b is a constant lim as x approaches c of b*f(x)= | b*lim as x approaches c of f(x) |
| lim as x approaches c of (f(x)+g(x)) = | lim as x approaches c of f(x) + lim as x approaches c of g(x) |
| lim as x approaches c of f(x)g(x)= | lim as x approaches c of f(x) * lim as x approaches c of g(x) |
| increasing exponentials dominate polynomials so | lim as x approaches c of polynomial/increasing exponential =0 |
| distance= | rate * time |
| for average velocity we care about | the direction of travel |
| for average speed we | ignore direction and only care about the total distance traveled |
| average velocity | the slope of the line segment connecting 2 points on the curve that is giving us height as a function of t |
| the average velocity of the object between t=a and t=b is | the change in position/ the change in time or (f(b)-f(a))/b-a |
| the instantaneous velocity of the object at time t=a is | lim as h approaches 0 of f(a+h)-f(a)/h |
| the average rate of change between x=a & x=a+h | f(a+h)-f(a)/h |
| to find the instantaneous rate of change | let a become smaller and smaller |
| f^1(a) is the | derivative of f at a |
| secant line | is the slope or average rate of change |
| instantaneous rate of change | is the slope of the tangent line to f at a |
| estimate lim as h approaches 0 of P(1+h)-P(1) by | plugging in h closer and closer to 0 |
| f^1(a) is | the rate at which the output of the function f(x) is changing at x=a |
| the derivative is | a function |
| the derivative of a function f^1 is given by | f^1(x)=lim as h approaches 0 of f(x+h)-f(x)/h |
| if the limit at all x exists | then f(x) is differentiable at all x |
| the domain of f^1(x) is | the x-values in the domain of f(x) so that the limit defining the derivative exists |
| if f^1(x)<0 on some interval | then f(x) is decreasing |
| if f^1(x)>0 on some interval | then f(x) is increasing |
| if f(x) is constant on some interval then | f^1(x)=0 for every point in the interval |
| if f(x) is linear on some interval then | f^1(x) is constant on that interval and equals the slope of the line |
| power rule | if f(x)=x^p then f^1(x)=px^p-1 |
| Leibniz notation | f^1(t)=dT/df |
| the derivative of a function at a point tells us | how the function is changing near that point |
| interpretations of the derivative must involve | a complete coherent sentence, the input and output is clear, the change in input is described, the direction of change is clear, indicates that it is an approximation, uses appropriate units |
Created by:
kzegelien2005
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