3.7 Implicit Differentiation
Example 1
Find
dy if y 2 = x using implicit differentiation. dx
What does this mean graphically? Consider the slope of the tangent lines at points (4, 2) and (4, 2), on 2 the curve y = x .
Example 2
Find the slope of the circl
AP Calculus AB A P C a l c u l u s B C
Free-Response Questions and Solutions 1989 1997
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Odds and Ends
Slope Fields Calculating Integrals from areas Average value of a function 2nd Fundamental Theorem More on the Integral as an Accumulator Differential equations revisited More Differential Equations and Exponential Growth and Decay More Tech
Calculus 12 AP Unit V Review of Big Ideas
Concept I Area
Region C Region A
b
Region B
d b
Region D
d
Area = f (x )dx
a
Area = f (y )dy
c
Area = ( f (x ) g( x )dx
a
Area = ( f (y) g(y)dy
c
General Comments Notice that a and b are on the x-axis c and d are
3.6 More on Chain Rule
Formulas for Derivatives of Composite Functions:
dn (u )= dx d (sin u ) = dx d (tan u) = dx d (sec u ) = dx d (cos u ) = dx d (cot u) = dx d (csc u) = dx
Example 1
An object moves along the x-axis so that its position at any time t
3.6 Chain Rule
General Discussion about how functions are formed
What is the difference between these functions:
y = x sin x
2
y=
x2 sin x
y = sin x
2
y = sin x
2
Example 1 Inside and outside functions
Identify the inside and the outside function in each
3.4 Velocity and Other Rates of Change
Remember slope of secant line
(diagram) distance traveled total time (assume you are going in one direction) Average velocity = What is this like?
slope of tangent line
Suppose you drive your car for 2 h and travel e
3.3 Rules for Differentiation
Basic Rules
Derivative of a Constant
(c ) =
Power Rule
(x ) =
n
Constant Multiple Rule
(c u) =
Sum and Difference Rule
(u v) =
Example 1 Differentiating using the basic rules
Find
dy if dx 5 a) y = x 3 + 6x 2 x + 16 3
b) y =
3.1
Derivative of a Function
Definition of the Derivative
Example 1 Applying the Definition
Differentiate (that is, find the derivative of) f (x ) = x
3
Calculus Section 2.1 Page1
Alternate Definition of the Derivative at a Point
Example 2 Applying the Al
CHAPTER 3 PRACTICE QUIZZES AND TESTS SOLUTIONS Quiz Section 3.1-3.3 1. 2. 3. 4. 5. 6. D GRAPH WITH A BUNCH OF POINTS! (sorry dont have my calculator on this computer) E C C B
Quiz Section 3.4-3.6 1. 2. 3. 4. 5. 6. 7. Piecewise graph with holes on each end
3.5 Derivatives of Trigonometric Functions
Derivative of the Sine Function
Sketch the y = sin x
2
2
3 2
2
2
3 2
2
2
Sketch the graph of the derivative of y = sin x on the grid below.
2
2
3 2
2
2
3 2
2
2
What do you notice? Now sketch the derivative
3.2 Differentiability
Things we discovered yesterday: f (x + h ) f ( x ) Definition of the Derivative: f ( x ) = lim h 0
h
The Derivative at a Point (2 formulas) f (a + h) f (a ) f (a ) = lim h 0 h Right-Hand Derivative at a f (a + h) f (a) f (a ) = lim h
Calculus 12AP
Three things to Review:
1. Solving Differential Equations Solve: dy x = dx y
Exponential Growth and Decay
2. Rate of Change
Class Notes
3. Direct Variation (grade 11) y varies directly as x _ * k is called the _ y varies inversely as x _
Rat
Applications of Integration Day 5
First idea: 2 Draw y = x on the axes below. Pick a point on the curve and label it: P (x, y). a) Draw the line y = 1 and find the distance d from the point P to the line
b) Draw the line y = 4 and find the distance d from
4.3b More Connecting f and f with the Graph of f
Look at Graph Summary Example 1 Using the First and Second Derivative
Draw a possible graph, given each set of conditions a) f (2) = 4, f (2 ) < 0, f (2) < 0 b) f (2) = 4, f (2 ) > 0, f (2) > 0
c) f (2) = 4
4.3 Connecting f and f with the Graph of f
First Derivative Test for Local Extrema
Theorem 4 First Derivative Test for Local Extrema The following test applies to a continuous function f (x ) . At a critical point c: 1. If f changes sign from positive to
4.2 Mean Value Theorem
ROLLE'S THEOREM Let f be differentiable on (a,b) and continuous on [a,b]. If f(a) = f(b) = 0, then there is at least one point c in (a,b) where f (c) = 0 . (geometrically obvious)
eg: (Set III #57) If c is the number defined by Roll
4.1 Extreme Values of Functions
Absolute (Global) Extreme Values
Example 1 Exploring Extreme Values
Determine the maximum value and minimum value on , of : 2 2 a) y = cos x b) y = sin x
Example 2 Exploring Absolute Extrema
Determine the absolute extreme
Unit II
The Derivative (22 days 1 test)
1. The Derivative of a Function
Definition of the Derivative as a limit of the difference quotient Alternate Definition of the Derivative at a Point Recognizing a given limit as a derivative Expressing the derivativ
3.9b More on Derivatives of Exponential and Logarithmic Functions
Yesterday you learned several new formulas (e x ) = (e u ) =
(ln x ) =
(ln u) =
The natural logarithm function obeys all of the laws of logarithms that you learned in Mathematics 12.
ln x =
3.9 Derivatives of Exponential and Logarithmic Fcns
Derivative of y = e
x
Graph Y1 = e x . Then graph the derivative of this function using the numerical derivative feature of your calculator: Y2 = nDeriv(Y1 , X , X )
What do you notice?
(e ) =
x
(e )
u
=