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symmetry about they-axis : the graph of a functionffis symmetric about theyy-axis if(âx,y)(âx,y)is on the graph offfwhenever(x,y)(x,y)is on the graph | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
table of values : a table containing a list of inputs and their corresponding outputs | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
transcendental function : a function that cannot be expressed by a combination of basic arithmetic operations | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
transformation of a function : a shift, scaling, or reflection of a function | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
trigonometric functions : functions of an angle defined as ratios of the lengths of the sides of a right triangle | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
trigonometric identity : an equation involving trigonometric functions that is true for all anglesθθfor which the functions in the equation are defined | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
vertical line test : given the graph of a function, every vertical line intersects the graph, at most, once | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
zeros of a function : when a real numberxxis a zero of a functionf,f(x)=0f,f(x)=0 | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
Differential calculus arose from trying to solve the problem of determining the slope of a line tangent to a curve at a point. The slope of the tangent line indicates the rate of change of the function, also called thederivative. Calculating a derivative requires finding a limit. | https://openstax.org/books/calculus-volume-1/pages/2-key-concepts |
Integral calculus arose from trying to solve the problem of finding the area of a region between the graph of a function and thex-axis. We can approximate the area by dividing it into thin rectangles and summing the areas of these rectangles. This summation leads to the value of a function called theintegral. The integ... | https://openstax.org/books/calculus-volume-1/pages/2-key-concepts |
Multivariable calculus enables us to solve problems in three-dimensional space, including determining motion in space and finding volumes of solids. | https://openstax.org/books/calculus-volume-1/pages/2-key-concepts |
A table of values or graph may be used to estimate a limit. | https://openstax.org/books/calculus-volume-1/pages/2-key-concepts |
If the limit of a function at a point does not exist, it is still possible that the limits from the left and right at that point may exist. | https://openstax.org/books/calculus-volume-1/pages/2-key-concepts |
If the limits of a function from the left and right exist and are equal, then the limit of the function is that common value. | https://openstax.org/books/calculus-volume-1/pages/2-key-concepts |
We may use limits to describe infinite behavior of a function at a point. | https://openstax.org/books/calculus-volume-1/pages/2-key-concepts |
The limit laws allow us to evaluate limits of functions without having to go through step-by-step processes each time. | https://openstax.org/books/calculus-volume-1/pages/2-key-concepts |
For polynomials and rational functions,limxâaf(x)=f(a).limxâaf(x)=f(a). | https://openstax.org/books/calculus-volume-1/pages/2-key-concepts |
You can evaluate the limit of a function by factoring and canceling, by multiplying by a conjugate, or by simplifying a complex fraction. | https://openstax.org/books/calculus-volume-1/pages/2-key-concepts |
The squeeze theorem allows you to find the limit of a function if the function is always greater than one function and less than another function with limits that are known. | https://openstax.org/books/calculus-volume-1/pages/2-key-concepts |
For a function to be continuous at a point, it must be defined at that point, its limit must exist at the point, and the value of the function at that point must equal the value of the limit at that point. | https://openstax.org/books/calculus-volume-1/pages/2-key-concepts |
Discontinuities may be classified as removable, jump, or infinite. | https://openstax.org/books/calculus-volume-1/pages/2-key-concepts |
A function is continuous over an open interval if it is continuous at every point in the interval. It is continuous over a closed interval if it is continuous at every point in its interior and is continuous at its endpoints. | https://openstax.org/books/calculus-volume-1/pages/2-key-concepts |
The composite function theorem states: Iff(x)f(x)is continuous atLandlimxâag(x)=L,limxâag(x)=L,thenlimxâaf(g(x))=f(limxâag(x))=f(L).limxâaf(g(x))=f(limxâag(x))=f(L). | https://openstax.org/books/calculus-volume-1/pages/2-key-concepts |
The Intermediate Value Theorem guarantees that if a function is continuous over a closed interval, then the function takes on every value between the values at its endpoints. | https://openstax.org/books/calculus-volume-1/pages/2-key-concepts |
The intuitive notion of a limit may be converted into a rigorous mathematical definition known as theepsilon-delta definition of the limit. | https://openstax.org/books/calculus-volume-1/pages/2-key-concepts |
The epsilon-delta definition may be used to prove statements about limits. | https://openstax.org/books/calculus-volume-1/pages/2-key-concepts |
The epsilon-delta definition of a limit may be modified to define one-sided limits. | https://openstax.org/books/calculus-volume-1/pages/2-key-concepts |
m sec = f ( x ) â f ( a ) x â a m sec = f ( x ) â f ( a ) x â a | https://openstax.org/books/calculus-volume-1/pages/2-key-equations |
v ave = s ( t ) â s ( a ) t â a v ave = s ( t ) â s ( a ) t â a | https://openstax.org/books/calculus-volume-1/pages/2-key-equations |
lim x â a f ( x ) = L lim x â a f ( x ) = L | https://openstax.org/books/calculus-volume-1/pages/2-key-equations |
lim x â a x = a lim x â a c = c lim x â a x = a lim x â a c = c | https://openstax.org/books/calculus-volume-1/pages/2-key-equations |
lim x â a â f ( x ) = L lim x â a + f ( x ) = L lim x â a â f ( x ) = L lim x â a + f ( x ) = L | https://openstax.org/books/calculus-volume-1/pages/2-key-equations |
lim x â a â f ( x ) = + â lim x â a â f ( x ) = â â lim x â a â f ( x ) = + â lim x â a â f ( x ) = â â | https://openstax.org/books/calculus-volume-1/pages/2-key-equations |
lim x â a + f ( x ) = + â lim x â a + f ( x ) = â â lim x â a + f ( x ) = + â lim x â a + f ( x ) = â â | https://openstax.org/books/calculus-volume-1/pages/2-key-equations |
lim x â a f ( x ) = + â : lim x â a â f ( x ) = + â lim x â a f ( x ) = + â : lim x â a â f ( x ) = + â and lim x â a + f ( x ) = + â lim x â a + f ( x ) = + â lim x â a f ( x ) = â â : lim x â a â f ( x ) = â â lim x â a f ( x ) = â â : lim x â a â f ( x ) = â â ... | https://openstax.org/books/calculus-volume-1/pages/2-key-equations |
lim x â a x = a lim x â a c = c lim x â a x = a lim x â a c = c | https://openstax.org/books/calculus-volume-1/pages/2-key-equations |
lim θ â 0 sin θ = 0 lim θ â 0 sin θ = 0 lim θ â 0 cos θ = 1 lim θ â 0 cos θ = 1 lim θ â 0 sin θ θ = 1 lim θ â 0 sin θ θ = 1 lim θ â 0 1 â cos θ θ = 0 lim θ â 0 1 â cos θ θ = 0 | https://openstax.org/books/calculus-volume-1/pages/2-key-equations |
average velocity : the change in an objectâs position divided by the length of a time period; the average velocity of an object over a time interval[t,a][t,a](ift<at<aor[a,t][a,t]ift>a)t>a), with a position given bys(t),s(t),that isvave=s(t)âs(a)tâavave=s(t)âs(a)tâa | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
constant multiple law for limits : the limit lawlimxâacf(x)=c·limxâaf(x)=cLlimxâacf(x)=c·limxâaf(x)=cL | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
continuity at a point : A functionf(x)f(x)is continuous at a pointaif and only if the following three conditions are satisfied: (1)f(a)f(a)is defined, (2)limxâaf(x)limxâaf(x)exists, and (3)limxâaf(x)=f(a)limxâaf(x)=f(a) | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
continuity from the left : A function is continuous from the left atbiflimxâbâf(x)=f(b)limxâbâf(x)=f(b) | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
continuity from the right : A function is continuous from the right ataiflimxâa+f(x)=f(a)limxâa+f(x)=f(a) | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
continuity over an interval : a function that can be traced with a pencil without lifting the pencil; a function is continuous over an open interval if it is continuous at every point in the interval; a functionf(x)f(x)is continuous over a closed interval of the form[a,b][a,b]if it is continuous at every point in(a,b),... | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
difference law for limits : the limit lawlimxâa(f(x)âg(x))=limxâaf(x)âlimxâag(x)=LâMlimxâa(f(x)âg(x))=limxâaf(x)âlimxâag(x)=LâM | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
differential calculus : the field of calculus concerned with the study of derivatives and their applications | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
discontinuity at a point : A function is discontinuous at a point or has a discontinuity at a point if it is not continuous at the point | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
epsilon-delta definition of the limit : limxâaf(x)=Llimxâaf(x)=Lif for everyε>0,ε>0,there exists aδ>0δ>0such that if0<|xâa|<δ,0<|xâa|<δ,then|f(x)âL|<ε|f(x)âL|<ε | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
infinite discontinuity : An infinite discontinuity occurs at a pointaiflimxâaâf(x)=±âlimxâaâf(x)=±âorlimxâa+f(x)=±âlimxâa+f(x)=±â | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
infinite limit : A function has an infinite limit at a pointaif it either increases or decreases without bound as it approachesa | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
instantaneous velocity : The instantaneous velocity of an object with a position function that is given bys(t)s(t)is the value that the average velocities on intervals of the form[t,a][t,a]and[a,t][a,t]approach as the values oftmove closer toa,a,provided such a value exists | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
integral calculus : the study of integrals and their applications | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
Intermediate Value Theorem : Letfbe continuous over a closed bounded interval[a,b];[a,b];ifzis any real number betweenf(a)f(a)andf(b),f(b),then there is a numbercin[a,b][a,b]satisfyingf(c)=zf(c)=z | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
intuitive definition of the limit : If all values of the functionf(x)f(x)approach the real numberLas the values ofx(âa)x(âa)approacha,f(x)f(x)approachesL | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
jump discontinuity : A jump discontinuity occurs at a pointaiflimxâaâf(x)limxâaâf(x)andlimxâa+f(x)limxâa+f(x)both exist, butlimxâaâf(x)âlimxâa+f(x)limxâaâf(x)âlimxâa+f(x) | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
limit : the process of lettingxortapproachain an expression; the limit of a functionf(x)f(x)asxapproachesais the value thatf(x)f(x)approaches asxapproachesa | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
limit laws : the individual properties of limits; for each of the individual laws, letf(x)f(x)andg(x)g(x)be defined for allxâaxâaover some open interval containinga; assume thatLandMare real numbers so thatlimxâaf(x)=Llimxâaf(x)=Landlimxâag(x)=M;limxâag(x)=M;letcbe a constant | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
multivariable calculus : the study of the calculus of functions of two or more variables | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
one-sided limit : A one-sided limit of a function is a limit taken from either the left or the right | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
power law for limits : the limit lawlimxâa(f(x))n=(limxâaf(x))n=Lnlimxâa(f(x))n=(limxâaf(x))n=Lnfor every positive integern | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
product law for limits : the limit lawlimxâa(f(x)·g(x))=limxâaf(x)·limxâag(x)=L·Mlimxâa(f(x)·g(x))=limxâaf(x)·limxâag(x)=L·M | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
quotient law for limits : the limit lawlimxâaf(x)g(x)=limxâaf(x)limxâag(x)=LMlimxâaf(x)g(x)=limxâaf(x)limxâag(x)=LMforMâ0Mâ0 | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
removable discontinuity : A removable discontinuity occurs at a pointaiff(x)f(x)is discontinuous ata, butlimxâaf(x)limxâaf(x)exists | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
root law for limits : the limit lawlimxâaf(x)n=limxâaf(x)n=Lnlimxâaf(x)n=limxâaf(x)n=Lnfor allLifnis odd and forLâ¥0Lâ¥0ifnis even | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
secant : A secant line to a functionf(x)f(x)atais a line through the point(a,f(a))(a,f(a))and another point on the function; the slope of the secant line is given bymsec=f(x)âf(a)xâamsec=f(x)âf(a)xâa | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
squeeze theorem : states that iff(x)â¤g(x)â¤h(x)f(x)â¤g(x)â¤h(x)for allxâaxâaover an open interval containingaandlimxâaf(x)=L=limxâah(x)limxâaf(x)=L=limxâah(x)whereLis a real number, thenlimxâag(x)=Llimxâag(x)=L | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
sum law for limits : The limit lawlimxâa(f(x)+g(x))=limxâaf(x)+limxâag(x)=L+Mlimxâa(f(x)+g(x))=limxâaf(x)+limxâag(x)=L+M | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
tangent : A tangent line to the graph of a function at a point(a,f(a))(a,f(a))is the line that secant lines through(a,f(a))(a,f(a))approach as they are taken through points on the function withx-values that approacha; the slope of the tangent line to a graph atameasures the rate of change of the function ata | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
triangle inequality : Ifaandbare any real numbers, then|a+b|â¤|a|+|b||a+b|â¤|a|+|b| | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
vertical asymptote : A function has a vertical asymptote atx=ax=aif the limit asxapproachesafrom the right or left is infinite | https://openstax.org/books/calculus-volume-1/pages/2-key-terms |
The slope of the tangent line to a curve measures the instantaneous rate of change of a curve. We can calculate it by finding the limit of the difference quotient or the difference quotient with incrementh.h. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
The derivative of a functionf(x)f(x)at a valueaais found using either of the definitions for the slope of the tangent line. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
Velocity is the rate of change of position. As such, the velocityv(t)v(t)at timettis the derivative of the positions(t)s(t)at timet.t.Average velocity is given byvave=s(t)âs(a)tâa.vave=s(t)âs(a)tâa.Instantaneous velocity is given byv(a)=sâ²(a)=limtâas(t)âs(a)tâa.v(a)=sâ²(a)=limtâas(t)âs(a)tâa. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
We may estimate a derivative by using a table of values. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
The derivative of a functionf(x)f(x)is the function whose value atxxisfâ²(x).fâ²(x). | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
The graph of a derivative of a functionf(x)f(x)is related to the graph off(x).f(x).Wheref(x)f(x)has a tangent line with positive slope,fâ²(x)>0.fâ²(x)>0.Wheref(x)f(x)has a tangent line with negative slope,fâ²(x)<0.fâ²(x)<0.Wheref(x)f(x)has a horizontal tangent line,fâ²(x)=0.fâ²(x)=0. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
If a function is differentiable at a point, then it is continuous at that point. A function is not differentiable at a point if it is not continuous at the point, if it has a vertical tangent line at the point, or if the graph has a sharp corner or cusp. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
Higher-order derivatives are derivatives of derivatives, from the second derivative to thenthnthderivative. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
The derivative of a constant function is zero. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
The derivative of a power function is a function in which the power onxxbecomes the coefficient of the term and the power onxxin the derivative decreases by 1. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
The derivative of a constantcmultiplied by a functionfis the same as the constant multiplied by the derivative. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
The derivative of the sum of a functionfand a functiongis the same as the sum of the derivative offand the derivative ofg. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
The derivative of the difference of a functionfand a functiongis the same as the difference of the derivative offand the derivative ofg. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
The derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
The derivative of the quotient of two functions is the derivative of the first function times the second function minus the derivative of the second function times the first function, all divided by the square of the second function. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
We used the limit definition of the derivative to develop formulas that allow us to find derivatives without resorting to the definition of the derivative. These formulas can be used singly or in combination with each other. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
Usingf(a+h)âf(a)+fâ²(a)h,f(a+h)âf(a)+fâ²(a)h,it is possible to estimatef(a+h)f(a+h)givenfâ²(a)fâ²(a)andf(a).f(a). | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
The rate of change of position is velocity, and the rate of change of velocity is acceleration. Speed is the absolute value, or magnitude, of velocity. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
The population growth rate and the present population can be used to predict the size of a future population. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
Marginal cost, marginal revenue, and marginal profit functions can be used to predict, respectively, the cost of producing one more item, the revenue obtained by selling one more item, and the profit obtained by producing and selling one more item. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
We can find the derivatives of sinxand cosxby using the definition of derivative and the limit formulas found earlier. The results areddxsinx=cosxddxcosx=âsinx.ddxsinx=cosxddxcosx=âsinx. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
With these two formulas, we can determine the derivatives of all six basic trigonometric functions. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
The chain rule allows us to differentiate compositions of two or more functions. It states that forh(x)=f(g(x)),h(x)=f(g(x)),hâ²(x)=fâ²(g(x))gâ²(x).hâ²(x)=fâ²(g(x))gâ²(x).In Leibnizâs notation this rule takes the formdydx=dydu·dudx.dydx=dydu·dudx. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
We can use the chain rule with other rules that we have learned, and we can derive formulas for some of them. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
The chain rule combines with the power rule to form a new rule:Ifh(x)=(g(x))n,thenhâ²(x)=n(g(x))nâ1gâ²(x).Ifh(x)=(g(x))n,thenhâ²(x)=n(g(x))nâ1gâ²(x). | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
When applied to the composition of three functions, the chain rule can be expressed as follows: Ifh(x)=f(g(k(x))),h(x)=f(g(k(x))),thenhâ²(x)=fâ²(g(k(x))gâ²(k(x))kâ²(x).hâ²(x)=fâ²(g(k(x))gâ²(k(x))kâ²(x). | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
The inverse function theorem allows us to compute derivatives of inverse functions without using the limit definition of the derivative. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
We can use the inverse function theorem to develop differentiation formulas for the inverse trigonometric functions. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
We use implicit differentiation to find derivatives of implicitly defined functions (functions defined by equations). | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
By using implicit differentiation, we can find the equation of a tangent line to the graph of a curve. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
On the basis of the assumption that the exponential functiony=bx,b>0y=bx,b>0is continuous everywhere and differentiable at 0, this function is differentiable everywhere and there is a formula for its derivative. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
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