Calculus Volume 1

# Key Concepts

Calculus Volume 1Key Concepts

### 4.1Related Rates

• To solve a related rates problem, first draw a picture that illustrates the relationship between the two or more related quantities that are changing with respect to time.
• In terms of the quantities, state the information given and the rate to be found.
• Find an equation relating the quantities.
• Use differentiation, applying the chain rule as necessary, to find an equation that relates the rates.
• Be sure not to substitute a variable quantity for one of the variables until after finding an equation relating the rates.

### 4.2Linear Approximations and Differentials

• A differentiable function $y=f(x)y=f(x)$ can be approximated at $aa$ by the linear function
$L(x)=f(a)+f′(a)(x−a).L(x)=f(a)+f′(a)(x−a).$
• For a function $y=f(x),y=f(x),$ if $xx$ changes from $aa$ to $a+dx,a+dx,$ then
$dy=f′(x)dxdy=f′(x)dx$

is an approximation for the change in $y.y.$ The actual change in $yy$ is
$Δy=f(a+dx)−f(a).Δy=f(a+dx)−f(a).$
• A measurement error $dxdx$ can lead to an error in a calculated quantity $f(x).f(x).$ The error in the calculated quantity is known as the propagated error. The propagated error can be estimated by
$dy≈f′(x)dx.dy≈f′(x)dx.$
• To estimate the relative error of a particular quantity $q,q,$ we estimate $Δqq.Δqq.$

### 4.3Maxima and Minima

• A function may have both an absolute maximum and an absolute minimum, have just one absolute extremum, or have no absolute maximum or absolute minimum.
• If a function has a local extremum, the point at which it occurs must be a critical point. However, a function need not have a local extremum at a critical point.
• A continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum. Each extremum occurs at a critical point or an endpoint.

### 4.4The Mean Value Theorem

• If $ff$ is continuous over $[a,b][a,b]$ and differentiable over $(a,b)(a,b)$ and $f(a)=0=f(b),f(a)=0=f(b),$ then there exists a point $c∈(a,b)c∈(a,b)$ such that $f′(c)=0.f′(c)=0.$ This is Rolle’s theorem.
• If $ff$ is continuous over $[a,b][a,b]$ and differentiable over $(a,b),(a,b),$ then there exists a point $c∈(a,b)c∈(a,b)$ such that
$f′(c)=f(b)−f(a)b−a.f′(c)=f(b)−f(a)b−a.$

This is the Mean Value Theorem.
• If $f′(x)=0f′(x)=0$ over an interval $I,I,$ then $ff$ is constant over $I.I.$
• If two differentiable functions $ff$ and $gg$ satisfy $f′(x)=g′(x)f′(x)=g′(x)$ over $I,I,$ then $f(x)=g(x)+Cf(x)=g(x)+C$ for some constant $C.C.$
• If $f′(x)>0f′(x)>0$ over an interval $I,I,$ then $ff$ is increasing over $I.I.$ If $f′(x)<0f′(x)<0$ over $I,I,$ then $ff$ is decreasing over $I.I.$

### 4.5Derivatives and the Shape of a Graph

• If $cc$ is a critical point of $ff$ and $f′(x)>0f′(x)>0$ for $x and $f′(x)<0f′(x)<0$ for $x>c,x>c,$ then $ff$ has a local maximum at $c.c.$
• If $cc$ is a critical point of $ff$ and $f′(x)<0f′(x)<0$ for $x and $f′(x)>0f′(x)>0$ for $x>c,x>c,$ then $ff$ has a local minimum at $c.c.$
• If $f″(x)>0f″(x)>0$ over an interval $I,I,$ then $ff$ is concave up over $I.I.$
• If $f″(x)<0f″(x)<0$ over an interval $I,I,$ then $ff$ is concave down over $I.I.$
• If $f′(c)=0f′(c)=0$ and $f″(c)>0,f″(c)>0,$ then $ff$ has a local minimum at $c.c.$
• If $f′(c)=0f′(c)=0$ and $f″(c)<0,f″(c)<0,$ then $ff$ has a local maximum at $c.c.$
• If $f′(c)=0f′(c)=0$ and $f″(c)=0,f″(c)=0,$ then evaluate $f′(x)f′(x)$ at a test point $xx$ to the left of $cc$ and a test point $xx$ to the right of $c,c,$ to determine whether $ff$ has a local extremum at $c.c.$

### 4.6Limits at Infinity and Asymptotes

• The limit of $f(x)f(x)$ is $LL$ as $x→∞x→∞$ (or as $x→−∞)x→−∞)$ if the values $f(x)f(x)$ become arbitrarily close to $LL$ as $xx$ becomes sufficiently large.
• The limit of $f(x)f(x)$ is $∞∞$ as $x→∞x→∞$ if $f(x)f(x)$ becomes arbitrarily large as $xx$ becomes sufficiently large. The limit of $f(x)f(x)$ is $−∞−∞$ as $x→∞x→∞$ if $f(x)<0f(x)<0$ and $|f(x)||f(x)|$ becomes arbitrarily large as $xx$ becomes sufficiently large. We can define the limit of $f(x)f(x)$ as $xx$ approaches $−∞−∞$ similarly.
• For a polynomial function $p(x)=anxn+an−1xn−1+…+a1x+a0,p(x)=anxn+an−1xn−1+…+a1x+a0,$ where $an≠0,an≠0,$ the end behavior is determined by the leading term $anxn.anxn.$ If $n≠0,n≠0,$ $p(x)p(x)$ approaches $∞∞$ or $−∞−∞$ at each end.
• For a rational function $f(x)=p(x)q(x),f(x)=p(x)q(x),$ the end behavior is determined by the relationship between the degree of $pp$ and the degree of $q.q.$ If the degree of $pp$ is less than the degree of $q,q,$ the line $y=0y=0$ is a horizontal asymptote for $f.f.$ If the degree of $pp$ is equal to the degree of $q,q,$ then the line $y=anbny=anbn$ is a horizontal asymptote, where $anan$ and $bnbn$ are the leading coefficients of $pp$ and $q,q,$ respectively. If the degree of $pp$ is greater than the degree of $q,q,$ then $ff$ approaches $∞∞$ or $−∞−∞$ at each end.

### 4.7Applied Optimization Problems

• To solve an optimization problem, begin by drawing a picture and introducing variables.
• Find an equation relating the variables.
• Find a function of one variable to describe the quantity that is to be minimized or maximized.
• Look for critical points to locate local extrema.

### 4.8L’Hôpital’s Rule

• L’Hôpital’s rule can be used to evaluate the limit of a quotient when the indeterminate form $0000$ or $∞/∞∞/∞$ arises.
• L’Hôpital’s rule can also be applied to other indeterminate forms if they can be rewritten in terms of a limit involving a quotient that has the indeterminate form $0000$ or $∞/∞.∞/∞.$
• The exponential function $exex$ grows faster than any power function $xp,xp,$ $p>0.p>0.$
• The logarithmic function $lnxlnx$ grows more slowly than any power function $xp,xp,$ $p>0.p>0.$

### 4.9Newton’s Method

• Newton’s method approximates roots of $f(x)=0f(x)=0$ by starting with an initial approximation $x0,x0,$ then uses tangent lines to the graph of $ff$ to create a sequence of approximations $x1,x2,x3,….x1,x2,x3,….$
• Typically, Newton’s method is an efficient method for finding a particular root. In certain cases, Newton’s method fails to work because the list of numbers $x0,x1,x2,…x0,x1,x2,…$ does not approach a finite value or it approaches a value other than the root sought.
• Any process in which a list of numbers $x0,x1,x2,…x0,x1,x2,…$ is generated by defining an initial number $x0x0$ and defining the subsequent numbers by the equation $xn=F(xn−1)xn=F(xn−1)$ for some function $FF$ is an iterative process. Newton’s method is an example of an iterative process, where the function $F(x)=x−[f(x)f′(x)]F(x)=x−[f(x)f′(x)]$ for a given function $f.f.$

### 4.10Antiderivatives

• If $FF$ is an antiderivative of $f,f,$ then every antiderivative of $ff$ is of the form $F(x)+CF(x)+C$ for some constant $C.C.$
• Solving the initial-value problem
$dydx=f(x),y(x0)=y0dydx=f(x),y(x0)=y0$

requires us first to find the set of antiderivatives of $ff$ and then to look for the particular antiderivative that also satisfies the initial condition.
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