Review Poster

1. Review Poster

2. 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

3. Number Line Analysis [instead of the big chart we used to make] (iii) No sign change at x = c, no local min/max

4. 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.

5. 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

6. 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 tangency • 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.

7. Properties of f(x) = ex • Inverse is y = lnx• • Domain (-∞,∞) • eaeb = ea+b • Range (0, ∞) • • • • elnx = x • ln ex = x

8. 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

9. Chain Rule (Composite Functions)

10. Rules for Differentiation Product rule: Quotient rule:

11. Implicit Differentiation When differentiating with respect to x (or t or θ) • 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 .

12. An Example of Implicit Differentiation Find if 2xy + y3+ x2 = 7 • 2y + 2x + 3y2 + 2x = 0 • (2x + 3y2)=-2x - 2y • =

13. 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.

14. 2x + 2y = 0 = -x/y = = = = (-y2 – x3 )/y3

15. 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 ___________ _____________)

16. Related Rates (The rates of change of two items are dependent) • Sketch • Identify what you know and what you want to find. • Write an equation. • Take the derivative of both sides of the equation. • Solve.

17. Big Section: Integrals • Approximate area under a curve • Riemann Sum = • Left endpoint • Right endpoint • Midpoint

18. 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

19. Circumscribed Rectangles -overestimate - happens when f is decreasing and you use left end point OR when f is increasing and you use right endpoint

20. 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

21. Fundamental Theorem of CalculusPart I If F’ = f,

22. Fundamental Theorem of CalculusPart II where a is a constant and x is a function

23. Properties of Definite Integrals

24. 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

25. Definition of Definite Integral

26. Average Value of a Function on [a,b] • M(x) = Average Value = • So, (b-a)(Avg. Val.)=

27. Total Distance Vs. Net Distance Net Distance over time [a,b] = Total Distance over time [a,b] =

28. 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 =

29. Volumes There 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

30. 3. Volume of a Known Cross-Section (foam projects) V = A(x) is the area of a known cross-section

31. You’re almost there! Only two more slides! (After this one) Next Big Section: Differential Equations and Slope Fields

32. Differential Equations • 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.

33. 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

34. The End