Calculus AB Topics Limits Continuity, Asymptotes
Consider f x 2x 1 x 3 1 x 3 x 3 Is there a vertical asymptote at x = 3? Do not give a Precalculus answer on a Calculus exam.
Consider f x 2x 1 x 3 1 x 3 x 3 Is there a horizontal asymptote at y = 0?
Definition of Continuity Continuity at an interior point lim x c f x f c Continuity at an endpoint - lim f x f a or lim f x f b x a x b
Derivative Highlights Limit Definitions of Derivative
Do you REALLY know the definitions of What are they??? a derivative??? f ( x) f ( a) lim x a x a lim ( ) ( ) f x h f x h 0 h
Example gx ( ) lim h 0 3 3 x h x h g(5)?
What does it mean for a function to be differentiable? The derivative exists at all x values or at a specified x value For a derivative to exist the LEFT hand derivative, must equal the RIGHT hand derivative. What is the example of a function that is continuous at a specific x value but not differentiable at that x-value?
product rule: d dv du uv u v dx dx dx Notice that this is not just the product of two derivatives. quotient rule: d u du dv v u dx dx dx v 2 v
Velocity = change in position change in time Average velocity = slope between two positions Instantaneous velocity = velocity at instant in time = derivative at a point Speed = velocity change in velocity Acceleration = change in time derivative of velocity = 2nd derivative of position
Summary of Trig Derivatives GOLDEN TICKET!!! d dx sin x cos x d dx csc x d dx cos x sin x d dx sec x d dx tan x d sec2 x dx cot x csc x cot x sec x tan x csc2 x Remember: sec 2 x sec x 2
Definition of the Chain Rule d dx f g x f ' g x g' x Derivative of outside function Derivative of inside function
Implicit Differentiation 1. x 2 y 2 25 Find dy dx 2x 2y dy dx 0 Power rule Chain rule Derivative of Constant is 0
Inverse Trig Rules GOLDEN TICKET!!! dy dx sin 1 u 1 1 u 2 du dy dx cos 1 u 1 1 u 2 du dy dx tan 1 u 1 1 u 2 du dy dx cot 1 u 1 1 u 2 du dy dx sec 1 u 1 u u 2 1 du dy dx csc 1 u 1 u u 2 1 du
Exponentials and Logs GOLDEN TICKET!!! d u u du d u u du e e a a ln a dx dx dx dx dy 1 1 ln u du dy loga ln u du dx u dx dx u ln a dx
Applications of Derivatives
Extrema and Concavity Critical points are where the derivative is zero or undefined. Extrema may occur at critical points and endpoints Finding extrema on a closed interval TEST the ENDPOINTS (Extreme Value Theorem) 1 st derivative tells you where the function is increasing/decreasing and possible extrema 2 nd derivative tells you where the function is concave up/down and possible points of inflection.
Sign charts Sign charts are valuable tools and are allowed, BUT THEY ARE NEVER NEVER NEVER SUFFICIENT TO EARN A POINT To earn the test points you must interpret the sign chart using words Your words should demonstrate that you understand the connection between the positive/negative behavior of a graph and the increasing/decreasing/extrema behavior of the parent function and how the increasing/decreasing behavior determines the extrema behavior
Intermediate Value Theorem If a function is continuous on [a,b] then the function takes on all values between f(a) and f(b).
Mean Value Theorem If f is continuous on [a,b] and differentiable on (a,b) Then there exists a number, c, in (a,b) such that f ' c f b b f a a
Optimization Write one or two equations that model the situation described. You may need to write two equations and use substitution. Take the derivative to find the maximum or minimum.
An open top box is to be made by cutting congruent squares from the corners of a 20-by-25 inch sheet of tin and bending up the sides. What dimensions give the box the LARGEST volume? Steps: 1. Set up an equation to maximize or minimize 1a. You may have 2 equations and substitute into one. 2. Take its derivative. 3. Set derivative = 0 and check slopes to show it is a max/min. 4. Answer the question asked.
Linearization A linearization is just another term for tangent line! You can use a linearization to estimate the value of a function at a given x-value. The closer the x-value is to the point of tangency the better the estimation.
Related Rates Draw a picture to model the problem Write an equation that reflects the model Pythagorean Theorem Trig Similar Figure (such as cones) Plug in values that never change Implicit differentiation Plug in values that do change and solve
Integration
Riemann Sums LRAM RRAM Midpoint - midpoint of x-interval, not the geometric midpoint Trapezoidal A 1 2 h b 0 2b 1 2b 2... 2b n 1 b n h width of partition b value of function at the specific x value If intervals are not uniform width, rectangles and trapezoids must be calculated individually
Fundamental Theorem of Calculus d dx F x d dx a x f t dt f x d dx u a f t dt f u du dx 5x d 3 3 Example t t dt x x dx 0 3 1 5 3 5 1 5
FTC Evaluative Part b a f x dx F b F a
Average Value Theorem: If a function is integrable on [a,b], then its average y-value is: Average b 1 a b a f x dx
5 x 2 2 5dx x2 5dx Is called a definite integral. We can evaluate it and get a numerical answer. Is called an indefinite integral. Its solution is the set of all possible antiderivatives.
Finding c Solve 3x 2 dx with the initial condition F(1) = 4 1. Find the antiderivative family 2. Substitute the given initial condition values 3. Solve for c 4. Substitute back into antiderivative
U - Substitution: 2 3 ( x 5) (2 x) dx
Inverse Trig Integrals 1. 4 4x 2 1 dx What inverse trig function looks like the 1 x 1 1 x 1 2 x 1 x 2 2 tan 1 sin 1 1 x sec x 1 c x c c best fit? Get the 1 in the bottom Factor constants outside the integral Write as (something) 2 Use u-substitution with inverse trig formula
Applications of Integrals
Velocity is the derivative of position. Position is the integral of velocity. Displacement is the distance from an arbitrary starting point at the end of some interval. It is the difference between ending position and starting position. Position Displacement v t Total distance is the how far an object travelled regardless of direction. b a dt Derivatives Velocity Acceleration Integrals Total Distance b a v t dt
Summary To find You Displacement b a v t dt Total Distance b a v t dt Position at a specific time Solve the indefinite integral and use given initial condition to find c v t dt P t c
Other Applications Velocities (distance and displacement) Rates of flow (e.g. oil leaking from a tanker or water flowing into a container) How many cars flow through an intersection How much money has accumulated in a bank account Anytime you know the rate of something and what to know how something has accumulated.
What if we want area between two curves? S b S top bottom a b S f x g x a dx
Why does x always have all the fun? Let s integrate with respect to y b dy a a b dx x value x value function in x dx y value y value function in y dy
Disks and Washers Disk method Washer method b r 2 dx b 2 2 R r dx a a
The axis of rotation is important Same equations = same area being rotated. But different axis of rotation = different solid