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

  is an approximation for the change in [latex]y[/latex]. The actual change in [latex]y[/latex] is

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

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

  This is the Mean Value Theorem.

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

Derivatives and the Shape of a Graph

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

Key Equations

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

Glossary

absolute extremum

if [latex]f[/latex] has an absolute maximum or absolute minimum at [latex]c[/latex], we say [latex]f[/latex] has an absolute extremum at [latex]c[/latex]

absolute maximum

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

absolute minimum

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

concave down

if [latex]f[/latex] is differentiable over an interval [latex]I[/latex] and [latex]f^{\prime}[/latex] is decreasing over [latex]I[/latex], then [latex]f[/latex] is concave down over [latex]I[/latex]

concave up

if [latex]f[/latex] is differentiable over an interval [latex]I[/latex] and [latex]f^{\prime}[/latex] is increasing over [latex]I[/latex], then [latex]f[/latex] is concave up over [latex]I[/latex]

concavity

the upward or downward curve of the graph of a function

concavity test

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

critical point

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

differential

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

differential form

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

extreme value theorem

if [latex]f[/latex] is a continuous function over a finite, closed interval, then [latex]f[/latex] has an absolute maximum and an absolute minimum

Fermat’s theorem

if [latex]f[/latex] has a local extremum at [latex]c[/latex], then [latex]c[/latex] is a critical point of [latex]f[/latex]

first derivative test

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

inflection point

if [latex]f[/latex] is continuous at [latex]c[/latex] and [latex]f[/latex] changes concavity at [latex]c[/latex], the point [latex](c,f(c))[/latex] is an inflection point of [latex]f[/latex]

linear approximation

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

local extremum

if [latex]f[/latex] has a local maximum or local minimum at [latex]c[/latex], we say [latex]f[/latex] has a local extremum at [latex]c[/latex]

local maximum

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

local minimum

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

mean value theorem

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

[latex]f^{\prime}(c)=\dfrac{f(b)-f(a)}{b-a}[/latex]

percentage error

the relative error expressed as a percentage

propagated error

the error that results in a calculated quantity [latex]f(x)[/latex] resulting from a measurement error [latex]dx[/latex]

relative error

given an absolute error [latex]\Delta q[/latex] for a particular quantity, [latex]\dfrac{\Delta q}{q}[/latex] 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 [latex]f[/latex] is continuous over [latex][a,b][/latex] and differentiable over [latex](a,b)[/latex], and if [latex]f(a)=f(b)[/latex], then there exists [latex]c \in (a,b)[/latex] such that [latex]f^{\prime}(c)=0[/latex]

second derivative test

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

tangent line approximation (linearization)

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

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 [latex]f(a)[/latex] is the [latex]y[/latex]-intercept and [latex]f'(a)[/latex] 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 ([latex]y[/latex]-value) and its location ([latex]x[/latex]-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 [latex]f'(x) = 0[/latex] and where [latex]f'(x)[/latex] 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 [latex]f'(x) = 0[/latex] and where [latex]f'(x)[/latex] 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 [latex]c[/latex], 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 [latex]f'(x)[/latex] and the graph of [latex]f(x)[/latex]
• Remember that not all critical points are local extrema
• Use a number line to organize information about [latex]f'(x)[/latex] signs

Concavity and Points of Inflection

• Practice finding [latex]f''(x)[/latex] for various function types
• Visualize how [latex]f''(x)[/latex] relates to the “bending” of the graph
• Remember that [latex]f''(x) = 0[/latex] doesn’t guarantee an inflection point
• Use a number line to organize information about [latex]f''(x)[/latex] 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