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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′(a)(x−a)
- For a function y=f(x), if x changes from a to a+dx, then
dy=f′(x)dx
is an approximation for the change in y. The actual change in y is
Δy=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≈f′(x)dx
- To estimate the relative error of a particular quantity q, we estimate Δqq
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∈(a,b) such that f′(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∈(a,b) such that
f′(c)=f(b)−f(a)b−a.
This is the Mean Value Theorem.
- If f′(x)=0 over an interval I, then f is constant over I.
- If two differentiable functions f and g satisfy f′(x)=g′(x) over I, then f(x)=g(x)+C for some constant C.
- If f′(x)>0 over an interval I, then f is increasing over I. If f′(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′(x)>0 for x<c and f′(x)<0 for x>c, then f has a local maximum at c.
- If c is a critical point of f and f′(x)<0 for x<c and f′(x)>0 for x>c, then f has a local minimum at c.
- If f′′(x)>0 over an interval I, then f is concave up over I.
- If f′′(x)<0 over an interval I, then f is concave down over I.
- If f′(c)=0 and f′′(c)>0, then f has a local minimum at c.
- If f′(c)=0 and f′′(c)<0, then f has a local maximum at c.
- If f′(c)=0 and f′′(c)=0, then evaluate f′(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′′(x)>0 over an interval I, then f is concave up over I.
- If f′′(x)<0 over an interval I, then f is concave down over I.
- If f′(c)=0 and f′′(c)>0, then f has a local minimum at c.
- If f′(c)=0 and f′′(c)<0, then f has a local maximum at c.
- If f′(c)=0 and f′′(c)=0, then evaluate f′(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′(a)(x−a) - A differential
dy=f′(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)≥f(x) for all x in the domain of f, we say f has an absolute maximum at c
- absolute minimum
- if f(c)≤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′ is decreasing over I, then f is concave down over I
- concave up
- if f is differentiable over an interval I and f′ 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′′>0 over I, then f is concave up over I; if f′′<0 over I, then f is concave down over I
- critical point
- if f′(c)=0 or f′(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′(x)dx
- differential form
- given a differentiable function y=f′(x), the equation dy=f′(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′ changes sign from positive to negative as x increases through c, then f has a local maximum at c; if f′ changes sign from negative to positive as x increases through c, then f has a local minimum at c; if f′ 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′(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)≥f(x) for all x∈I, we say f has a local maximum at c
- local minimum
- if there exists an interval I such that f(c)≤f(x) for all x∈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∈(a,b) such that
f′(c)=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 Δq for a particular quantity, Δqq 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∈(a,b) such that f′(c)=0
- second derivative test
- suppose f′(c)=0 and f′′ is continuous over an interval containing c; if f′′(c)>0, then f has a local minimum at c; if f′′(c)<0, then f has a local maximum at c; if f′′(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