review poster
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Review Poster. First Derivative Test [Local min/max]. If x = c is a critical value on f and f’ changes sign at x = c… ( i ) f has a local max at x = c if f’ changes from >0 to 0. - PowerPoint PPT PresentationTRANSCRIPT
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Review Poster
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First Derivative Test [Local min/max]
If x = c is a critical value on f and f’ changes sign at x = c…
(i) f has a local max at x = c if f’ changes from >0 to <0
(ii) f has a local min at x = c if f’ changes from < 0 to > 0
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Number Line Analysis [instead of the big chart we used to
make]
(iii) No sign change at x = c, no local min/max
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Second Derivative Test [Also a test for local min/max, not a test for concavity or points of inflection]
If x = c is a critical value on f [meaning f’(c) = 0 or is undefined] and f”(c) exists…
(i) If f”(c) > 0, x = c is a local min on f(ii) If f”(c) < 0, x = c is a local max on f(iii) If f”(c) = 0, then the test fails and we don’t
know anything.
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Test for Concavity on f [Points of Inflection]
Evaluate f” at points where f’=0•Point of inflection on f at x = c, f changes from
concave down to concave up
• Point of inflection on f at x = c, f changes from concave up to concave down
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Linear Approximation
Use equation of a line tangent to f at a point (x, f(x)) to estimate values of f(x) close to the
point of tangency1) If f is concave up (f” > 0), then the linear approximation will
be less than the true value
2) If f is concave down (f”< 0), then the linear approximation will be greater than the true value.
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Properties of f(x) = ex
• Inverse is y = lnx • • Domain (-∞,∞) • ea eb = ea+b
• Range (0, ∞) • • • • elnx = x
• ln ex = x
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Properties of f(x) = lnx• Inverse is y = ex • ln(ab) = lna + lnb• Domain (0, ∞) • ln (a/b) = lna - lnb• Range (- ∞, ∞) • ln(ak) = klna
• Always concave • ln x <0 if 0<x<1 down• Reflection over x-axis: -lnx• reflect over y-axis: ln(-x) • horizon. shift a units • vert. shift a units left: ln(x + a) up: ln(x) + a• horizon. shift a units • vert. shift a units right: ln(x – a) down: ln (x) - a
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Chain Rule (Composite Functions)
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Rules for Differentiation
Product rule:
Quotient rule:
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Implicit Differentiation
When differentiating with respect to x (or t or θ)1) Differentiate both sides with respect to x, t,
or θ.2) Collect all terms with on one side of the
equation.3) Factor out . 4) Solve for .
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An Example of Implicit Differentiation
Find if 2xy + y3+ x2 = 7
• 2y + 2x + 3y2 + 2x = 0
• (2x + 3y2)=-2x - 2y
• =
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Another (slightly different) example of implicit differentiation
If x2 + y2 = 10, find .
I will pause here to let you catch up on copying and try to solve this problem on your own.
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2x + 2y = 0
= -x/y
=
= = = (-y2 – x3 )/y3
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Line Tangent to Curve at a Point
- Need slope (derivative) at a point (original function)
- A line normal to a curve at a point is ______________ to the tangent line at that point. (The slopes of these lines will be ___________ _____________)
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Related Rates (The rates of change of two items are dependent)
1. Sketch2. Identify what you know and what you want
to find.3. Write an equation.4. Take the derivative of both sides of the
equation.5. Solve.
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Big Section: Integrals
- Approximate area under a curve- Riemann Sum =- Left endpoint- Right endpoint- Midpoint
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Inscribed and Circumscribed Rectangles
Inscribed Rectangles - underestimate - happens when f is decreasing and you use
right end point OR when f is increasing and you use left endpoint
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• Circumscribed Rectangles-overestimate
- happens when f is decreasing and you use left end point OR when f is increasing and you use right endpoint
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Trapezoidal Rule – most accurate approximation
If f is continuous on [a,b]
•As n ∞, this estimate is extremely accurate• Trapezoidal rule is always the average of left
and right Riemann sums
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Fundamental Theorem of CalculusPart I
If F’ = f,
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Fundamental Theorem of CalculusPart II
where a is a constant and x is a function
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Properties of Definite Integrals
• • • •
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Oh man. This is taking me so long to type. So. Long.
• If f is an odd function (symmetric about the origin, (a,b) (-a,-b)) , then
• If f is an even function (symmetric about the y-
axis, (a,b) (-a,b))), then • If f(x) ≥ g(x) on [a,b] then
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Definition of Definite Integral
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Average Value of a Function on [a,b]
• M(x) = Average Value =• • So, (b-a)(Avg. Val.)=
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Total Distance Vs. Net Distance
Net Distance over time [a,b] =
Total Distance over time [a,b] =
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Area Between Two Curves
If f(x) and g(x) are continuous on [a,b] such that f(x) g(x), then the area between f(x) and g(x) is given by
Area =
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VolumesThere are basically three types of volume problems…
1. Volume by Rotation – Disc - (x-section is a circle)
(where R is the radius from the axis of rev.)
2. Volume by Rotation – Washer -(x-section is a circle)
R(x) = radius from axis of revolution to outer
figure r(x) = radius from axis of revolution to
inner figure
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3. Volume of a Known Cross-Section (foam projects)
V =
A(x) is the area of a known cross-section
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You’re almost there! Only two more slides! (After this one)
Next Big Section: Differential Equations and Slope Fields
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Differential Equations
1. Separate dy and dx algebraically. [Separation of variables.]
2. both sides. This will create a c value. The general solution has a c in it.
3. Solve for c using the initial conditions. Use this c value to write the particular solution.
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Slope Fields-Show a graphical solution to differential equations- Big picture made of tangent segments is the
solution- The slope of each individual tangent segment is
the value of at that point
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The End