AP Calculus AB‎ > ‎Previous Units‎ > ‎

04 &06 - Integration & Inverses



Assessment Standards

4.3 Can apply the Fundamental Theorem of Calculus to compute integrals.
6.2 Can compute the derivatives and integrals involving exponential functions.
6.4 Can compute the derivative and integrals involving log functions.



 

Chapter 4 - Integrals

 

Areas and Distances

·    Computation of Riemann sums using left, right, and midpoint evaluation.

If we know how something is changing, how can we find out how much the function has changed in total?

Riemann Sums

HW: Using rectangles to approximate area under a curve

HW: Riemann sums and sigma notation


The Definite Integral

·    Definite integral as limit of Riemann sums over equal subdivisions.

·    Basic properties of definite integral, such as additivity, and linearity.

·    Use of Riemann sums to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values.

Just as we used the limit to precisely study and define the derivative, we’ll now use it to precisely define the integral – a function that tells us how much something has changed.

Riemann sums and integrals

HW: The definite integral as the limit of a Riemann sum

Properties of Definite Integrals

KHW: Properties of Integrals

The Fundamental Theorem of Calculus

·    The FTC, using the integral of a rate of change to give accumulated change. Use of FTC to evaluate definite integrals.

·    Use of FTC to represent a particular anti-derivative and the analytic and graphical analysis of functions so defined.

How are the derivative and the integral related?


Functions defined by integrals

The Fundamental Theorem of Calculus

KHW: Functions defined as Integrals

KHW: Antiderivatives

KHW: FTC

Indefinite Integrals and the Net Change Theorem

·    Finding specific anti-derivatives using initial conditions, including applications to motion along a line.

·    Interpretation of FTC2 as the total change of a rate function.

·    Representation of an indefinite integral as a family of functions.

Using integrals, we can now find out exactly how much a physical system has changed over time. We’ll see how the initial values of variables describing a system can impact how the system changes over time.

Indefinite Integrals and anti-derivatives

KHW: Indefinite Integrals

The Substitution Rule

·    Anti-derivatives by substitution of variables, including change of limits for definite integrals.

·    The two methods of using substitution to compute definite integrals.

Real world functions are often complicated. We will learn how to simplify functions using substitution, and see how this impacts the system graphically.

The Reverse Chain Rule

Back Substitution

U-Subs with Definite Integrals



We're working through the un-starred units in chapter 6.

Chapter 6 - Logarithmic and Exponential Functions

 

Inverse Functions

·    The use of multiple representations (verbal, numeric, visual, algebraic) to understand inverse functions, always returning to the central idea of reversing inputs and outputs.

·    Use of implicit differentiation to find the derivative of an inverse function.

Functions relate inputs to outputs. What if we’re given the output and we need to find the input? Most functions have inverse functions, and we’ll learn how inverse functions are treated in calculus.

Inverse Functions

Non-Invertible Functions


Exponential Functions and Their Derivatives

·    Comparing relative magnitudes of functions, their rates of change (e.g. exponential vs. polynomial vs. logarithmic).

·    Algebraic and geometric properties of exponential functions.

·    Translation and reflection of exponential functions, from both symbolic and geometric perspectives.

·    Derivatives of exponential functions. The definition of e.

Exponential functions are everywhere in nature, from biology, to nuclear physics. We need to understand their behavior, and how to study the rate of change of exponential functions.

Exponential Functions

The number e

The Natural Exponential Function

Calculus of General Exponentials Functions

Logarithmic Functions

·    Comparing magnitudes of functions, their rates of change.

·    Logarithmic functions and their properties, including their geometric properties as inverses of exponentials.

·    Graphs of logarithmic functions, including asymptotic behavior.

The logarithm is the inverse of the exponential. What are logarithmic functions, and why are they useful?

Logarithmic Functions

Solving Exponential and Logarithmic Functions

Changing the Base of a Logarithmic Function


Derivatives of Logarithmic Functions

·    Derivatives and integrals of logarithmic functions.


We will discover the huge importance of the derivative of logarithm functions.

Natural Logarithmic Functions and their Derivatives

Calculus of General Logarithmic Functions

Derivatives of Inverse Trig Functions

·    Derivatives of asin(x), acos(x) and atan(x)


Derivatives of Inverse Trig Functions

Antiderivatives involving Inverse Functions





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4A - Practice, Practice, Practice! [u4.1: 1, 3, 5, 7, 15, 17, 19] [u4.2: 1, 5, 7, 9, 17, 23, 29, 33, 35, 37, 47, 55, 59] 
4B - yet more practice! [u4.3: 1-47 odd] 
4C [u4.4: 5, 9, 11, 15, 17, 21, 33, 35, 39, 45, 47, 57] 
4D [u4.5: 1-10, 11, 13, 17, 21, 23, 31, 39, 47, 53] 
6A - Inverse Functions and Exponentials [u6.1: 1, 3, 5, 9, 13, 17, 19, 21, 23, 25, 31, 35, 39, 41] [u6.2: 1, 5, 11, 15, 25, 29, 33, 37, 43, 51, 67, 69, 79, 83, 87] 
6B - Logarithms and Derivatives of Logs [u6.3: 1, 5, 7, 11, 13, 15, 21, 25, 29, 35, 37, 39, 47, 49, 57, 59, 61, 65] [u6.4: 7, 9, 15, 19, 27, 33, 35, 47, 73, 75, 81] 
6C - Inverse Trig Functions [u6.6: 1, 3, 5, 7, 13, 15, 23, 29, 31, 37, 43, 45, 59, 61, 63] 
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Nikhil Joshi,
Dec 5, 2017, 11:59 AM
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Nikhil Joshi,
Dec 13, 2017, 12:07 PM
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Nikhil Joshi,
Dec 4, 2017, 10:49 AM