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absolute maximum : iff(c)â¥f(x)f(c)â¥f(x)for allxxin the domain off,f,we sayffhas an absolute maximum atcc | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
absolute minimum : iff(c)â¤f(x)f(c)â¤f(x)for allxxin the domain off,f,we sayffhas an absolute minimum atcc | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
antiderivative : a functionFFsuch thatFâ²(x)=f(x)Fâ²(x)=f(x)for allxxin the domain offfis an antiderivative offf | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
concave down : ifffis differentiable over an intervalIIandfâ²fâ²is decreasing overI,I,thenffis concave down overII | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
concave up : ifffis differentiable over an intervalIIandfâ²fâ²is increasing overI,I,thenffis concave up overII | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
concavity : the upward or downward curve of the graph of a function | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
concavity test : supposeffis twice differentiable over an intervalI;I;iffâ³>0fâ³>0overI,I,thenffis concave up overI;I;iffâ³<0fâ³<0overI,I,thenffis concave down overII | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
critical number : iffâ²(c)=0fâ²(c)=0orfâ²(c)fâ²(c)is undefined, we say thatccis a critical number offf | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
critical point : the point(c,f(c))(c,f(c))a critical point offf | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
differential : the differentialdxdxis an independent variable that can be assigned any nonzero real number; the differentialdydyis defined to bedy=fâ²(x)dxdy=fâ²(x)dx | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
differential form : given a differentiable functiony=fâ²(x),y=fâ²(x),the equationdy=fâ²(x)dxdy=fâ²(x)dxis the differential form of the derivative ofyywith respect toxx | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
end behavior : the behavior of a function asxââxââandxâââxâââ | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
extreme value theorem : ifffis a continuous function over a finite, closed interval, thenffhas an absolute maximum and an absolute minimum | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
Fermatâs theorem : ifffhas a local extremum atc,c,thenccis a critical point offf | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
first derivative test : letffbe a continuous function over an intervalIIcontaining a critical pointccsuch thatffis differentiable overIIexcept possibly atc;c;iffâ²fâ²changes sign from positive to negative asxxincreases throughc,c,thenffhas a local maximum atc;c;iffâ²fâ²changes sign from negative to positive asxxinc... | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
horizontal asymptote : iflimxââf(x)=Llimxââf(x)=Lorlimxâââf(x)=L,limxâââf(x)=L,theny=Ly=Lis a horizontal asymptote offf | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
indefinite integral : the most general antiderivative off(x)f(x)is the indefinite integral off;f;we use the notationâ«f(x)dxâ«f(x)dxto denote the indefinite integral offf | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
indeterminate forms : when evaluating a limit, the forms00,00,â/â,â/â,0·â,0·â,âââ,âââ,00,00,â0,â0,and1â1âare considered indeterminate because further analysis is required to determine whether the limit exists and, if so, what its value is | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
infinite limit at infinity : a function that becomes arbitrarily large asxbecomes large | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
inflection point : ifffis continuous atccandffchanges concavity atc,c,the point(c,f(c))(c,f(c))is an inflection point offf | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
initial value problem : a problem that requires finding a functionyythat satisfies the differential equationdydx=f(x)dydx=f(x)together with the initial conditiony(x0)=y0y(x0)=y0 | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
iterative process : process in which a list of numbersx0,x1,x2,x3â¦x0,x1,x2,x3â¦is generated by starting with a numberx0x0and definingxn=F(xnâ1)xn=F(xnâ1)fornâ¥1nâ¥1 | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
LâHôpitalâs rule : ifffandggare differentiable functions over an intervala,a,except possibly ata,a,andlimxâaf(x)=0=limxâag(x)limxâaf(x)=0=limxâag(x)orlimxâaf(x)limxâaf(x)andlimxâag(x)limxâag(x)are infinite, thenlimxâaf(x)g(x)=limxâafâ²(x)gâ²(x),limxâaf(x)g(x)=limxâafâ²(x)gâ²(x),assuming... | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
limit at infinity : the limiting value, if it exists, of a function asxââxââorxâââxâââ | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
linear approximation : the linear functionL(x)=f(a)+fâ²(a)(xâa)L(x)=f(a)+fâ²(a)(xâa)is the linear approximation offfatx=ax=a | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
local extremum : ifffhas a local maximum or local minimum atc,c,we sayffhas a local extremum atcc | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
local maximum : if there exists an intervalIIsuch thatf(c)â¥f(x)f(c)â¥f(x)for allxâI,xâI,we sayffhas a local maximum atcc | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
local minimum : if there exists an intervalIIsuch thatf(c)â¤f(x)f(c)â¤f(x)for allxâI,xâI,we sayffhas a local minimum atcc | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
mean value theorem : ifffis continuous over[a,b][a,b]and differentiable over(a,b),(a,b),then there existscâ(a,b)câ(a,b)such thatfâ²(c)=f(b)âf(a)bâafâ²(c)=f(b)âf(a)bâa | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
Newtonâs method : method for approximating roots off(x)=0;f(x)=0;using an initial guessx0;x0;each subsequent approximation is defined by the equationxn=xnâ1âf(xnâ1)fâ²(xnâ1)xn=xnâ1âf(xnâ1)fâ²(xnâ1) | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
oblique asymptote : the liney=mx+by=mx+biff(x)f(x)approaches it asxââxââorxâââxâââ | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
optimization problems : problems that are solved by finding the maximum or minimum value of a function | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
percentage error : the relative error expressed as a percentage | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
propagated error : the error that results in a calculated quantityf(x)f(x)resulting from a measurement errordx | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
related rates : are rates of change associated with two or more related quantities that are changing over time | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
relative error : given an absolute errorÎqÎqfor a particular quantity,ÎqqÎqqis the relative error. | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
rolleâs theorem : ifffis continuous over[a,b][a,b]and differentiable over(a,b),(a,b),and iff(a)=f(b),f(a)=f(b),then there existscâ(a,b)câ(a,b)such thatfâ²(c)=0fâ²(c)=0 | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
second derivative test : supposefâ²(c)=0fâ²(c)=0andfâ³fâ³is continuous over an interval containingc;c;iffâ³(c)>0,fâ³(c)>0,thenffhas a local minimum atc;c;iffâ³(c)<0,fâ³(c)<0,thenffhas a local maximum atc;c;iffâ³(c)=0,fâ³(c)=0,then the test is inconclusive | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
tangent line approximation (linearization) : since the linear approximation offfatx=ax=ais defined using the equation of the tangent line, the linear approximation offfatx=ax=ais also known as the tangent line approximation toffatx=ax=a | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
The use of sigma (summation) notation of the formâi=1naiâi=1naiis useful for expressing long sums of values in compact form. | https://openstax.org/books/calculus-volume-1/pages/5-key-concepts |
For a continuous function defined over an interval[a,b],[a,b],the process of dividing the interval intonequal parts, extending a rectangle to the graph of the function, calculating the areas of the series of rectangles, and then summing the areas yields an approximation of the area of that region. | https://openstax.org/books/calculus-volume-1/pages/5-key-concepts |
The width of each rectangle isÎx=bâan.Îx=bâan. | https://openstax.org/books/calculus-volume-1/pages/5-key-concepts |
Riemann sums are expressions of the formâi=1nf(xi*)Îx,âi=1nf(xi*)Îx,and can be used to estimate the area under the curvey=f(x).y=f(x).Left- and right-endpoint approximations are special kinds of Riemann sums where the values of{xi*}{xi*}are chosen to be the left or right endpoints of the subintervals, respectivel... | https://openstax.org/books/calculus-volume-1/pages/5-key-concepts |
Riemann sums allow for much flexibility in choosing the set of points{xi*}{xi*}at which the function is evaluated, often with an eye to obtaining a lower sum or an upper sum. | https://openstax.org/books/calculus-volume-1/pages/5-key-concepts |
The definite integral can be used to calculate net signed area, which is the area above thex-axis less the area below thex-axis. Net signed area can be positive, negative, or zero. | https://openstax.org/books/calculus-volume-1/pages/5-key-concepts |
The component parts of the definite integral are the integrand, the variable of integration, and the limits of integration. | https://openstax.org/books/calculus-volume-1/pages/5-key-concepts |
Continuous functions on a closed interval are integrable. Functions that are not continuous may still be integrable, depending on the nature of the discontinuities. | https://openstax.org/books/calculus-volume-1/pages/5-key-concepts |
The properties of definite integrals can be used to evaluate integrals. | https://openstax.org/books/calculus-volume-1/pages/5-key-concepts |
The area under the curve of many functions can be calculated using geometric formulas. | https://openstax.org/books/calculus-volume-1/pages/5-key-concepts |
The average value of a function can be calculated using definite integrals. | https://openstax.org/books/calculus-volume-1/pages/5-key-concepts |
The Mean Value Theorem for Integrals states that for a continuous function over a closed interval, there is a valuecsuch thatf(c)f(c)equals the average value of the function. SeeThe Mean Value Theorem for Integrals. | https://openstax.org/books/calculus-volume-1/pages/5-key-concepts |
The Fundamental Theorem of Calculus, Part 1 shows the relationship between the derivative and the integral. SeeFundamental Theorem of Calculus, Part 1. | https://openstax.org/books/calculus-volume-1/pages/5-key-concepts |
The Fundamental Theorem of Calculus, Part 2 is a formula for evaluating a definite integral in terms of an antiderivative of its integrand. The total area under a curve can be found using this formula. SeeThe Fundamental Theorem of Calculus, Part 2. | https://openstax.org/books/calculus-volume-1/pages/5-key-concepts |
The net change theorem states that when a quantity changes, the final value equals the initial value plus the integral of the rate of change. Net change can be a positive number, a negative number, or zero. | https://openstax.org/books/calculus-volume-1/pages/5-key-concepts |
The area under an even function over a symmetric interval can be calculated by doubling the area over the positivex-axis. For an odd function, the integral over a symmetric interval equals zero, because half the area is negative. | https://openstax.org/books/calculus-volume-1/pages/5-key-concepts |
Substitution is a technique that simplifies the integration of functions that are the result of a chain-rule derivative. The term âsubstitutionâ refers to changing variables or substituting the variableuanddufor appropriate expressions in the integrand. | https://openstax.org/books/calculus-volume-1/pages/5-key-concepts |
When using substitution for a definite integral, we also have to change the limits of integration. | https://openstax.org/books/calculus-volume-1/pages/5-key-concepts |
Exponential and logarithmic functions arise in many real-world applications, especially those involving growth and decay. | https://openstax.org/books/calculus-volume-1/pages/5-key-concepts |
Substitution is often used to evaluate integrals involving exponential functions or logarithms. | https://openstax.org/books/calculus-volume-1/pages/5-key-concepts |
Formulas for derivatives of inverse trigonometric functions developed inDerivatives of Exponential and Logarithmic Functionslead directly to integration formulas involving inverse trigonometric functions. | https://openstax.org/books/calculus-volume-1/pages/5-key-concepts |
Use the formulas listed in the rule on integration formulas resulting in inverse trigonometric functions to match up the correct format and make alterations as necessary to solve the problem. | https://openstax.org/books/calculus-volume-1/pages/5-key-concepts |
Substitution is often required to put the integrand in the correct form. | https://openstax.org/books/calculus-volume-1/pages/5-key-concepts |
â i = 1 n c = n c â i = 1 n c = n c â i = 1 n c a i = c â i = 1 n a i â i = 1 n c a i = c â i = 1 n a i â i = 1 n ( a i + b i ) = â i = 1 n a i + â i = 1 n b i â i = 1 n ( a i + b i ) = â i = 1 n a i + â i = 1 n b i â i = 1 n ( a i â b i ) = â i = 1 n a i â â i = 1 n b i â i = 1 n ( ... | https://openstax.org/books/calculus-volume-1/pages/5-key-equations |
â i = 1 n i = 1 + 2 + ⯠+ n = n ( n + 1 ) 2 â i = 1 n i = 1 + 2 + ⯠+ n = n ( n + 1 ) 2 â i = 1 n i 2 = 1 2 + 2 2 + ⯠+ n 2 = n ( n + 1 ) ( 2 n + 1 ) 6 â i = 1 n i 2 = 1 2 + 2 2 + ⯠+ n 2 = n ( n + 1 ) ( 2 n + 1 ) 6 â i = 0 n i 3 = 1 3 + 2 3 + ⯠+ n 3 = n 2 ( n + 1 ) 2 4 â i = 0 n i 3 = 1 3 + 2 3 ... | https://openstax.org/books/calculus-volume-1/pages/5-key-equations |
A â L n = f ( x 0 ) Î x + f ( x 1 ) Î x + ⯠+ f ( x n â 1 ) Î x = â i = 1 n f ( x i â 1 ) Î x A â L n = f ( x 0 ) Î x + f ( x 1 ) Î x + ⯠+ f ( x n â 1 ) Î x = â i = 1 n f ( x i â 1 ) Î x | https://openstax.org/books/calculus-volume-1/pages/5-key-equations |
A â R n = f ( x 1 ) Î x + f ( x 2 ) Î x + ⯠+ f ( x n ) Î x = â i = 1 n f ( x i ) Î x A â R n = f ( x 1 ) Î x + f ( x 2 ) Î x + ⯠+ f ( x n ) Î x = â i = 1 n f ( x i ) Î x | https://openstax.org/books/calculus-volume-1/pages/5-key-equations |
â« a b f ( x ) d x = lim n â â â i = 1 n f ( x i * ) Î x â« a b f ( x ) d x = lim n â â â i = 1 n f ( x i * ) Î x | https://openstax.org/books/calculus-volume-1/pages/5-key-equations |
â« a a f ( x ) d x = 0 â« a a f ( x ) d x = 0 â« b a f ( x ) d x = â â« a b f ( x ) d x â« b a f ( x ) d x = â â« a b f ( x ) d x â« a b [ f ( x ) + g ( x ) ] d x = â« a b f ( x ) d x + â« a b g ( x ) d x â« a b [ f ( x ) + g ( x ) ] d x = â« a b f ( x ) d x + â« a b g ( x ) d x â« a b [ f ( x ) â g ... | https://openstax.org/books/calculus-volume-1/pages/5-key-equations |
If f ( x ) f ( x ) is continuous over an interval [ a , b ] , [ a , b ] , then there is at least one point c â [ a , b ] c â [ a , b ] such that f ( c ) = 1 b â a â« a b f ( x ) d x . f ( c ) = 1 b â a â« a b f ( x ) d x . | https://openstax.org/books/calculus-volume-1/pages/5-key-equations |
If f ( x ) f ( x ) is continuous over an interval [ a , b ] , [ a , b ] , and the function F ( x ) F ( x ) is defined by F ( x ) = â« a x f ( t ) d t , F ( x ) = â« a x f ( t ) d t , then F â² ( x ) = f ( x ) . F â² ( x ) = f ( x ) . | https://openstax.org/books/calculus-volume-1/pages/5-key-equations |
If f is continuous over the interval [ a , b ] [ a , b ] and F ( x ) F ( x ) is any antiderivative of f ( x ) , f ( x ) , then â« a b f ( x ) d x = F ( b ) â F ( a ) . â« a b f ( x ) d x = F ( b ) â F ( a ) . | https://openstax.org/books/calculus-volume-1/pages/5-key-equations |
F ( b ) = F ( a ) + â« a b F ' ( x ) d x F ( b ) = F ( a ) + â« a b F ' ( x ) d x or â« a b F ' ( x ) d x = F ( b ) â F ( a ) â« a b F ' ( x ) d x = F ( b ) â F ( a ) | https://openstax.org/books/calculus-volume-1/pages/5-key-equations |
â« f [ g ( x ) ] g â² ( x ) d x = â« f ( u ) d u = F ( u ) + C = F ( g ( x ) ) + C â« f [ g ( x ) ] g â² ( x ) d x = â« f ( u ) d u = F ( u ) + C = F ( g ( x ) ) + C | https://openstax.org/books/calculus-volume-1/pages/5-key-equations |
â« a b f ( g ( x ) ) g ' ( x ) d x = â« g ( a ) g ( b ) f ( u ) d u â« a b f ( g ( x ) ) g ' ( x ) d x = â« g ( a ) g ( b ) f ( u ) d u | https://openstax.org/books/calculus-volume-1/pages/5-key-equations |
â« e x d x = e x + C â« e x d x = e x + C â« a x d x = a x ln a + C â« a x d x = a x ln a + C | https://openstax.org/books/calculus-volume-1/pages/5-key-equations |
â« x â1 d x = ln | x | + C â« x â1 d x = ln | x | + C â« ln x d x = x ln x â x + C = x ( ln x â 1 ) + C â« ln x d x = x ln x â x + C = x ( ln x â 1 ) + C â« log a x d x = x ln a ( ln x â 1 ) + C â« log a x d x = x ln a ( ln x â 1 ) + C | https://openstax.org/books/calculus-volume-1/pages/5-key-equations |
â« d u a 2 â u 2 = sin â1 ( u a ) + C â« d u a 2 â u 2 = sin â1 ( u a ) + C â« d u a 2 + u 2 = 1 a tan â1 ( u a ) + C â« d u a 2 + u 2 = 1 a tan â1 ( u a ) + C â« d u u u 2 â a 2 = 1 a sec â1 ( u a ) + C â« d u u u 2 â a 2 = 1 a sec â1 ( u a ) + C | https://openstax.org/books/calculus-volume-1/pages/5-key-equations |
average value of a function : (orfave) the average value of a function on an interval can be found by calculating the definite integral of the function and dividing that value by the length of the interval | https://openstax.org/books/calculus-volume-1/pages/5-key-terms |
change of variables : the substitution of a variable, such asu, for an expression in the integrand | https://openstax.org/books/calculus-volume-1/pages/5-key-terms |
definite integral : a primary operation of calculus; the area between the curve and thex-axis over a given interval is a definite integral | https://openstax.org/books/calculus-volume-1/pages/5-key-terms |
fundamental theorem of calculus : the theorem, central to the entire development of calculus, that establishes the relationship between differentiation and integration | https://openstax.org/books/calculus-volume-1/pages/5-key-terms |
fundamental theorem of calculus, part 1 : uses a definite integral to define an antiderivative of a function | https://openstax.org/books/calculus-volume-1/pages/5-key-terms |
fundamental theorem of calculus, part 2 : (also,evaluation theorem) we can evaluate a definite integral by evaluating the antiderivative of the integrand at the endpoints of the interval and subtracting | https://openstax.org/books/calculus-volume-1/pages/5-key-terms |
integrable function : a function is integrable if the limit defining the integral exists; in other words, if the limit of the Riemann sums asngoes to infinity exists | https://openstax.org/books/calculus-volume-1/pages/5-key-terms |
integrand : the function to the right of the integration symbol; the integrand includes the function being integrated | https://openstax.org/books/calculus-volume-1/pages/5-key-terms |
integration by substitution : a technique for integration that allows integration of functions that are the result of a chain-rule derivative | https://openstax.org/books/calculus-volume-1/pages/5-key-terms |
left-endpoint approximation : an approximation of the area under a curve computed by using the left endpoint of each subinterval to calculate the height of the vertical sides of each rectangle | https://openstax.org/books/calculus-volume-1/pages/5-key-terms |
limits of integration : these values appear near the top and bottom of the integral sign and define the interval over which the function should be integrated | https://openstax.org/books/calculus-volume-1/pages/5-key-terms |
lower sum : a sum obtained by using the minimum value off(x)f(x)on each subinterval | https://openstax.org/books/calculus-volume-1/pages/5-key-terms |
mean value theorem for integrals : guarantees that a pointcexists such thatf(c)f(c)is equal to the average value of the function | https://openstax.org/books/calculus-volume-1/pages/5-key-terms |
net change theorem : if we know the rate of change of a quantity, the net change theorem says the future quantity is equal to the initial quantity plus the integral of the rate of change of the quantity | https://openstax.org/books/calculus-volume-1/pages/5-key-terms |
net signed area : the area between a function and thex-axis such that the area below thex-axis is subtracted from the area above thex-axis; the result is the same as the definite integral of the function | https://openstax.org/books/calculus-volume-1/pages/5-key-terms |
partition : a set of points that divides an interval into subintervals | https://openstax.org/books/calculus-volume-1/pages/5-key-terms |
regular partition : a partition in which the subintervals all have the same width | https://openstax.org/books/calculus-volume-1/pages/5-key-terms |
riemann sum : an estimate of the area under the curve of the formAââi=1nf(xi*)ÎxAââi=1nf(xi*)Îx | https://openstax.org/books/calculus-volume-1/pages/5-key-terms |
right-endpoint approximation : the right-endpoint approximation is an approximation of the area of the rectangles under a curve using the right endpoint of each subinterval to construct the vertical sides of each rectangle | https://openstax.org/books/calculus-volume-1/pages/5-key-terms |
sigma notation : (also,summation notation) the Greek letter sigma (Σ) indicates addition of the values; the values of the index above and below the sigma indicate where to begin the summation and where to end it | https://openstax.org/books/calculus-volume-1/pages/5-key-terms |
total area : total area between a function and thex-axis is calculated by adding the area above thex-axis and the area below thex-axis; the result is the same as the definite integral of the absolute value of the function | https://openstax.org/books/calculus-volume-1/pages/5-key-terms |
upper sum : a sum obtained by using the maximum value off(x)f(x)on each subinterval | https://openstax.org/books/calculus-volume-1/pages/5-key-terms |
variable of integration : indicates which variable you are integrating with respect to; if it isx, then the function in the integrand is followed bydx | https://openstax.org/books/calculus-volume-1/pages/5-key-terms |
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