Using TI-83/84 CALCULATE functions

Calc BC Review Material (primarily Chapters 1-10)AP Calculus BC

Make sure you practice with the functions in red To Display the CALCULATE menu, press 2nd-CALC 1: value Calculate a function Y value for a given X Finds a zero (x-intercept) of a function Part B #82 2: zero 3: minimum Finds a minimum of a function 4: maximum Finds a maximum of a function 5: intersect Finds an intersection of two functions 6: dy/dx Finds a numeric derivative of a function Part B #80 7: f ( x )dx Finds a numeric integral of a function

References are from AP Calc 2003 BC Exam, Parts A & B

Unit Circle: (x,y) = (cos, sin)Must know trig functions of common angles!!

2.1: Differentiability and Continuity Theorem 2.1 Differentiability Implies Continuity If f is differentiable at x = c, then f is continuous at x = c Question: Does continuity imply differentiability? i.e. If a function is continuous at x=c, does that mean it is also differentiable at x=c? Answer : NO!!

AP Exam Part A - #13, Part B - #76

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2.2/2.3 Differentiation Rules The Power Rule (Theorem 2.3) Product Rule Quotient Rule Trig Functions Exponential

2.2: Position, Velocity, Acceleration Generally position is s(t), x(t), h(t) position at time t Velocity is derivative of the position function Acceleration is derivative of velocity To find min/max Position examine its derivative (set ds/dt=0) Velocity examine its derivative (set dv/dt=0) Acceleration examine its derivative (set da/dt=0)Read question carefully what is it asking for?

d n [u ] = nu n 1u ' dxd [uv ] = uv '+ vu ' dxd u vu ' uv ' = dx v v2 d [sin x] = cos x dxd [cos x] = sin x dx

d [tan x] = sec2 x dxd [sec x] = sec x tan x dx

d [cot x] = csc2 x dx d [csc x] = csc x cot x dx

d u e = eu u ' dx

d u' [ln u ] = dx u

AP Exam Part B - #87, #91

2.4: Differentiation - Chain Rule One of the most powerful differentiation rules! NEED TO KNOW for the AP exam! Deals with composite functionsOuter function

2.5: Implicit Differentiation For functions not defined in the explicit y=f(x) form do implicit differentiation Differentiate both sides of eqn with respect to x Collect all terms involving dy/dx on left side Move all other terms to right side Solve for dy/dx Example: 2 2

y = f ( g ( x )) = f (u )Inner Function

x + 4 y = 7 + 3 xy

AP Exam Part A #1, #3, #8, #9,

AP Exam Part B #79

2004 FRQ #4 (a)

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2.6: Guidelines for Solving Related Rates dxdt

3.2: Mean Value Theorem for Derivatives

6 mi

s

1.

x

Identify all given quantities and quantities to be determined. (Make a sketch) 2. Write an equation for any given rates of change. State which rates of change we are to find. 3. Implicitly differentiate both sides of the equation with respect to time t. 4. Substitute into equation all known values for variables and their rates of change. Solve for the required rates of change. AP Exam Part B - #78 AP Exam Part A - #12

Instantaneous Rate of Change

Average Rate of Change

AP Calc AB Exam Part B - #83, 92

1.4 Intermediate Value Theorem If the function f(x) is continuous on [a, b], and y is a number between f(a) and f(b), then there exists at least one number x = c in the open interval (a, b) such that f(c) = y..

Curve Sketching and Analysisy = f(x) must be continuous at each: dy critical point: dx = 0 or undefined. And dont forget endpoints local minimum: local maximum:dy dx

3.1 Extreme Value Theorem.

goes (,0,+) or (,und,+) or dx 2 > 0d2y dx 2

d2y

If f(x) is continuous on [a, b], then f has both a minimum and a maximum on the interval.AP Calc AB Exam Part B - #80

dy goes (+,0,) or (+,und,) or dx

B=-1 Let x=3 => A=1 x 3dx + x 2dx1 1

4) Integrate

AP Exam Part A #26

Can only apply to indeterminate forms f(x) and g(x) are differentiable and g(x) is not 0

LHopitals RuleAP 2003 Exam Part A #2

8.8 : A Special Type of Improper Integral

lim

e x cos x 2 x x 0 x2 2 x

Indeterminate form?If so, can apply LHopitals: lim f(x)/g(x) = lim f(x)/g(x) To apply LHopitals repeatedly, remember to check if indeterminate at each stage

Remember p-series? This is the integral version of the pseries! Same convergence test converges when p>1.

AP Exam Part A - #6

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9.2: Geometric Series vs. P-Series

9.7: Definition of nth degree Taylor and Maclaurin Polynomial

Converges if |r| < 1 NOTE: a/xn is geo series where r=(1/x) AP Exam Part A #9 Converges if p > 1 Divergences if 0 divergent harmonic series If p=1 and numerator=(-1)n -> alternating harmonic series

AP Exam Part A #22 (along with Comparison Test), #24 (geo,p-series, nth term)

AP 2006 FRQ #6, AP 2005 FRQ #6(a,b), 2004 FRQ #6(a,b,d)

9.10: Taylor and MacLaurin Series

10.3: Finding Slope of a Tangent Line to a Curve defined by Parametric Equations

Given series defn for f(x), to find f(g(x)), just substitute g(x) for x in the power series expansion for f(x). To find f(x),simply differentiate each term, including the general term AP Exam Part A #11 AP Exam Part A - #20

Finding slope of tangent line AP Exam Part A #4 AP 2006 FRQ #3(c) Finding eqn of tangent line AP Exam Part A #17, AP 2004 FRQ #3 (b)

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10.3: Finding Higher-Order Derivatives

Parametric Equations and VectorValued Functions Parametric equations x(t) and y(t) can also be represented as a vector r(t)= r(t) represents the position vector at time t

Velocity = v(t)=r(t)= Acceleration = a(t)=r(t)= Speed of the particle is the magnitude of the velocity vector 2 2|| v(t) || = || r(t) || = dx dy + dt dt

Finding acceleration vector

AP 2006 FRQ #3 (a)

Represented as - see next slide

Speed at given time t AP Exam Part B - #84, AP2006 FRQ #3 Find t when v(t)= AP Exam Part A - #7

AP Tips re: Parametric Eqns For particles whose position is portrayed via parametric equations When is the particle at rest? When dy/dt=0 and dx/dt=0 (i.e. not moving in any direction)

10.4: Graphing with Polar Coordinates Area (sector) =1 2 r 2

When does the path of the particle have a horizontal tangent? When dy/dt= 0 (i.e. particle not moving in y direction)

P

When does the path of the particle have a vertical tangent? When dx/dt = 0 (i.e. particle not moving in x direction)

2005 AP FRQ #2(a) area in polar coords #2(b) conversion between rectangular and polar coordinates #2(c) decreasing polar function (negative 1st derivative) #2(d) 1st derivative test (max r occurs when dr/d0 = 0)

If dy/dt > 0, particle is moving upwards (dy/dt 0, particle is moving to the right (dx/dt A Y STO> B Dont use decimal approximation for pi !!!! i.e. use the key on the calculator, dont approximate to 3.14 for example(unless you want to type in minimum 10 decimal places for pi!)

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