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Essential Concepts

Related 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.

Linear Approximations and Differentials

• A differentiable function $y=f(x)$ can be approximated at $a$ by the linear function
  $L(x)=f(a)+f^{\prime}(a)(x-a)$
• For a function $y=f(x)$, if $x$ changes from $a$ to $a+dx$, then
  $dy=f^{\prime}(x) \, dx$

  is an approximation for the change in $y$. The actual change in $y$ is

  $\Delta y=f(a+dx)-f(a)$
• A measurement error $dx$ can lead to an error in a calculated quantity $f(x)$. The error in the calculated quantity is known as the propagated error. The propagated error can be estimated by
  $dy\approx f^{\prime}(x) \, dx$
• To estimate the relative error of a particular quantity $q$, we estimate $\dfrac{\Delta q}{q}$

Maxima 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.

The Mean Value Theorem

• If $f$ is continuous over $[a,b]$ and differentiable over $(a,b)$ and $f(a)=0=f(b)$, then there exists a point $c \in (a,b)$ such that $f^{\prime}(c)=0$. This is Rolle’s theorem.
• If $f$ is continuous over $[a,b]$ and differentiable over $(a,b)$, then there exists a point $c \in (a,b)$ such that
  $f^{\prime}(c)=\dfrac{f(b)-f(a)}{b-a}$.

  This is the Mean Value Theorem.

• If $f^{\prime}(x)=0$ over an interval $I$, then $f$ is constant over $I$.
• If two differentiable functions $f$ and $g$ satisfy $f^{\prime}(x)=g^{\prime}(x)$ over $I$, then $f(x)=g(x)+C$ for some constant $C$.
• If $f^{\prime}(x)>0$ over an interval $I$, then $f$ is increasing over $I$. If $f^{\prime}(x)<0$ over $I$, then $f$ is decreasing over $I$.

Derivatives and the Shape of a Graph

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

Key Equations

• Linear approximation
  $L(x)=f(a)+f^{\prime}(a)(x-a)$
• A differential
  $dy=f^{\prime}(x) \, dx$.

Glossary

absolute extremum

if $f$ has an absolute maximum or absolute minimum at $c$, we say $f$ has an absolute extremum at $c$

absolute maximum

if $f(c)\ge f(x)$ for all $x$ in the domain of $f$, we say $f$ has an absolute maximum at $c$

absolute minimum

if $f(c)\le f(x)$ for all $x$ in the domain of $f$, we say $f$ has an absolute minimum at $c$

concave down

if $f$ is differentiable over an interval $I$ and $f^{\prime}$ is decreasing over $I$, then $f$ is concave down over $I$

concave up

if $f$ is differentiable over an interval $I$ and $f^{\prime}$ is increasing over $I$, then $f$ is concave up over $I$

concavity

the upward or downward curve of the graph of a function

concavity test

suppose $f$ is twice differentiable over an interval $I$; if $f^{\prime \prime}>0$ over $I$, then $f$ is concave up over $I$; if $f^{\prime \prime}<0$ over $I$, then $f$ is concave down over $I$

critical point

if $f^{\prime}(c)=0$ or $f^{\prime}(c)$ is undefined, we say that $c$ is a critical point of $f$

differential

the differential $dx$ is an independent variable that can be assigned any nonzero real number; the differential $dy$ is defined to be $dy=f^{\prime}(x) \, dx$

differential form

given a differentiable function $y=f^{\prime}(x)$, the equation $dy=f^{\prime}(x) \, dx$ is the differential form of the derivative of $y$ with respect to $x$

extreme value theorem

if $f$ is a continuous function over a finite, closed interval, then $f$ has an absolute maximum and an absolute minimum

Fermat’s theorem

if $f$ has a local extremum at $c$, then $c$ is a critical point of $f$

first derivative test

let $f$ be a continuous function over an interval $I$ containing a critical point $c$ such that $f$ is differentiable over $I$ except possibly at $c$; if $f^{\prime}$ changes sign from positive to negative as $x$ increases through $c$, then $f$ has a local maximum at $c$; if $f^{\prime}$ changes sign from negative to positive as $x$ increases through $c$, then $f$ has a local minimum at $c$; if $f^{\prime}$ does not change sign as $x$ increases through $c$, then $f$ does not have a local extremum at $c$

inflection point

if $f$ is continuous at $c$ and $f$ changes concavity at $c$, the point $(c,f(c))$ is an inflection point of $f$

linear approximation

the linear function $L(x)=f(a)+f^{\prime}(a)(x-a)$ is the linear approximation of $f$ at $x=a$

local extremum

if $f$ has a local maximum or local minimum at $c$, we say $f$ has a local extremum at $c$

local maximum

if there exists an interval $I$ such that $f(c)\ge f(x)$ for all $x\in I$, we say $f$ has a local maximum at $c$

local minimum

if there exists an interval $I$ such that $f(c)\le f(x)$ for all $x\in I$, we say $f$ has a local minimum at $c$

mean value theorem

if $f$ is continuous over $[a,b]$ and differentiable over $(a,b)$, then there exists $c \in (a,b)$ such that

$f^{\prime}(c)=\dfrac{f(b)-f(a)}{b-a}$

percentage error

the relative error expressed as a percentage

propagated error

the error that results in a calculated quantity $f(x)$ resulting from a measurement error $dx$

relative error

given an absolute error $\Delta q$ for a particular quantity, $\dfrac{\Delta q}{q}$ is the relative error.

related rates

are rates of change associated with two or more related quantities that are changing over time

rolle’s theorem

if $f$ is continuous over $[a,b]$ and differentiable over $(a,b)$, and if $f(a)=f(b)$, then there exists $c \in (a,b)$ such that $f^{\prime}(c)=0$

second derivative test

suppose $f^{\prime}(c)=0$ and $f^{\prime \prime}$ is continuous over an interval containing $c$; if $f^{\prime \prime}(c)>0$, then $f$ has a local minimum at $c$; if $f^{\prime \prime}(c)<0$, then $f$ has a local maximum at $c$; if $f^{\prime \prime}(c)=0$, then the test is inconclusive

tangent line approximation (linearization)

since the linear approximation of $f$ at $x=a$ is defined using the equation of the tangent line, the linear approximation of $f$ at $x=a$ is also known as the tangent line approximation to $f$ at $x=a$

Study Tips

Related-Rates Problem-Solving

• Practice identifying the related quantities in word problems.
• Memorize common geometric formulas (area, volume, Pythagorean theorem).
• Review chain rule applications, as they’re crucial in related rates.
• Always check if your answer makes physical sense in the context of the problem.
• Be careful with positive/negative rates (e.g., increasing vs. decreasing quantities).

Linear Approximation of a Function at a Point

• Visualize the tangent line and the function to understand the approximation geometrically.
• Compare approximations with actual function values to gauge accuracy.
• Remember that $f(a)$ is the $y$-intercept and $f'(a)$ is the slope of the approximation line.
• Use linear approximation to estimate values of complex functions or irrational numbers.

Differentials and Amount of Error

• Practice computing differentials for various functions.
• Understand the connection between differentials and the slope of the tangent line.
• In error estimation problems, identify the measured quantity and the calculated quantity.
• Remember that relative error is often more meaningful than absolute error.
• Practice converting between absolute, relative, and percentage errors.
• When estimating errors, consider both positive and negative variations in measurements.

Absolute Extrema

• Practice identifying absolute extrema on various function graphs.
• Understand the difference between the extremum ($y$-value) and its location ($x$-value).
• Recognize scenarios where the Extreme Value Theorem applies or doesn’t apply.
• Pay attention to the domain of the function, especially its endpoints and continuity.
• Learn to distinguish between functions that have both, one, or no absolute extrema.

Local Extrema and Critical Points

• Practice identifying local extrema on various function graphs.
• Learn to distinguish between local and absolute extrema.
• When finding critical points, check both for $f'(x) = 0$ and where $f'(x)$ is undefined.
• Remember that critical points are only candidates for local extrema, not guarantees.
• Pay attention to endpoints of closed intervals, as they may contain extrema but are not considered local extrema.

Locating Absolute Extrema

• Always check both endpoints of the interval, even if they’re not critical points.
• When finding critical points, check for both $f'(x) = 0$ and where $f'(x)$ is undefined.
• Don’t forget to evaluate the function at each critical point within the interval.
• Organize your work: list all candidate points (endpoints and critical points) with their function values.
• Remember that a function can have multiple points with the same extreme value.

Rolle’s Theorem

• Verify all conditions of Rolle’s Theorem before applying it.
• Practice identifying functions that satisfy or violate the conditions.
• Use Rolle’s Theorem to prove the existence of zeros of a derivative, not to find them explicitly.

The Mean Value Theorem and Its Meaning

• Visualize the theorem geometrically: think of a secant line and parallel tangent line(s).
• Connect the Mean Value Theorem with the concept of average rate of change.
• Use the theorem to solve problems involving motion, particularly relating average and instantaneous velocities.
• Remember that the theorem doesn’t tell you how to find $c$, just that it exists.

Corollaries of the Mean Value Theorem

• Understand the relationship between these corollaries and the Mean Value Theorem.
• Practice applying each corollary to different types of functions and scenarios.
• For Corollary 1, remember that it applies to an entire interval, not just a single point.
• For Corollary 2, think about how this relates to the concept of antiderivatives.
• For Corollary 3, visualize how the sign of the derivative affects the graph of the function.

The First Derivative Test

• Practice finding critical points for various function types
• Visualize the relationship between $f'(x)$ and the graph of $f(x)$
• Remember that not all critical points are local extrema
• Use a number line to organize information about $f'(x)$ signs

Concavity and Points of Inflection

• Practice finding $f''(x)$ for various function types
• Visualize how $f''(x)$ relates to the “bending” of the graph
• Remember that $f''(x) = 0$ doesn’t guarantee an inflection point
• Use a number line to organize information about $f''(x)$ signs
• Compare concavity analysis with first derivative analysis for a complete picture
• Sketch graphs to confirm your analytical results

The Second Derivative Test

• Use this test in conjunction with other analytical tools (e.g., concavity analysis)
• Be prepared to use the First Derivative Test when this test is inconclusive