Presentation on theme: "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."— Presentation transcript:
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
Number Line Analysis [instead of the big chart we used to make] (iii) No sign change at x = c, no local min/max
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.
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
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 1)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.
Properties of f(x) = e x Inverse is y = lnx Domain (-∞,∞) e a e b = e a+b Range (0, ∞) e lnx = x ln e x = x
Properties of f(x) = lnx Inverse is y = e x ln(ab) = lna + lnb Domain (0, ∞) ln (a/b) = lna - lnb Range (- ∞, ∞) ln(a k ) = klna Always concave ln x <0 if 0
Chain Rule (Composite Functions)
Rules for Differentiation Product rule: Quotient rule:
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.
An Example of Implicit Differentiation Find if 2xy + y 3 + x 2 = 7 2y + 2x + 3y 2 + 2x = 0 (2x + 3y 2 )=-2x - 2y =
Another (slightly different) example of implicit differentiation If x 2 + y 2 = 10, find. I will pause here to let you catch up on copying and try to solve this problem on your own.
2x + 2y = 0 = -x/y = = = (-y 2 – x 3 )/y 3
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 ___________ _____________)
Related Rates (The rates of change of two items are dependent) 1.Sketch 2.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.
Big Section: Integrals -Approximate area under a curve -Riemann Sum = -Left endpoint -Right endpoint -Midpoint
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
Circumscribed Rectangles -overestimate - happens when f is decreasing and you use left end point OR when f is increasing and you use right endpoint
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
Fundamental Theorem of Calculus Part I If F’ = f,
Fundamental Theorem of Calculus Part II where a is a constant and x is a function
Properties of Definite Integrals
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
Definition of Definite Integral
Average Value of a Function on [a,b] M(x) = Average Value = So, (b-a)(Avg. Val.)=
Total Distance Vs. Net Distance Net Distance over time [a,b] = Total Distance over time [a,b] =
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 =
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
3. Volume of a Known Cross-Section (foam projects) V = A(x) is the area of a known cross-section
You’re almost there! Only two more slides! (After this one) Next Big Section: Differential Equations and Slope Fields
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.
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