# AP CALCULUS AB ACADEMIC LIBRARY

*Second Semester*

## GETTING STARTED IN EDGENUITY

#### THE STUDENT EXPERIENCE

#### MATH COURSE

## AP Calculus AB Second Semester

**Unit 6: Definite Integrals**

**Introduction to Unit 6****Instruction – Unit 6 Introduction**

**Estimating with Finite Sums****Instruction – Learn about estimating with finite sums.****Example Problem: Using Approximation Methods. How are the approximation methods used to estimate area under a curve?****Example Problem: Relating Distance to a Velocity vs. Time Graph. How does the area under the curve apply to velocity?****Example Problem: Approximating the Amount Accumulated. How do you estimate the amount accumulated?****Example Problem: Connecting the Area under the Curve to the Volume of a Sphere. How can you extend the area under a curve to estimate the volume of a sphere?****Example Problem: Estimating Population Density. How do you apply approximations to determine population density?**

**Reading Lesson 6.1: Estimating with Finite Sums****Instruction – Reading Lesson****Practice Problems 6.1 – Read section 6.1 in the textbook and complete associated exercises.**

**Definite Integrals****Instruction – Learn about definite integrals.****Example Problem: Expressing Riemann Sums as Definite Integrals. How do you use integral notation to express a limit of Riemann sums?****Example Problem: Using Area to Evaluate Definite Integrals. How do you evaluate definite integrals using area formulas?****Example Problem: Writing Accumulator Functions. How can you use definite integrals to solve problems using accumulation?****Example Problem: Evaluating Definite Integrals of Non-Continuous Functions. How do you evaluate definite integrals with discontinuities?**

**Reading Lesson 6.2: Definite Integrals****Instruction – Reading Lesson****Practice Problems 6.2 – Read section 6.2 in the textbook and complete associated exercises.**

**Definite Integrals and Antiderivatives****Instruction – Learn about definite integrals and antiderivatives.****Example Problem: Using the Properties of Definite Integrals. What are the properties of definite integrals?****Example Problem: Using the Properties of Definite Integrals. How can you calculate definite integrals using their properties?****Example Problem: Using the Properties of Definite Integrals to Evaluate the Upper and Lower Bounds for an Integral. How can you evaluate the upper and lower bounds of a definite integral?****Example Problem: Using the Mean Value Theorem. How can you use the mean value theorem for definite integrals to find the point where the average value occurs?****Example Problem: Calculating the Integral with Antiderivatives. How is the area under the curve related to antiderivatives?**

**Reading Lesson 6.3: Definite Integrals and Antiderivatives****Instruction – Reading Lesson****Practice Problems 6.3 – Read section 6.3 in the textbook and complete associated exercises.**

**Fundamental Theorem of Calculus, Parts 1 and 2****Instruction – Learn about the fundamental theorem of calculus.****Example Problem: Applying the First Part of the Fundamental Theorem. How do you determine derivatives of functions defined with integrals?****Example Problem: Analyzing Graphs with the Fundamental Theorem of Calculus. How do you analyze graphs with the fundamental theorem of calculus?****Example Problem: Constructing a Function with the Fundamental Theorem. How can you use the fundamental theorem of calculus to build a function?****Example Problem: Applying the Second Part of the Fundamental Theorem. How do you evaluate definite integrals with the fundamental theorem of calculus?****Example Problem: Analyzing Rates of Change in the Real World. How do you analyze rates of change with the fundamental theorem of calculus?**

**Reading Lesson 6.4: Fundamental Theorem of Calculus****Instruction – Reading Lesson****Practice Problems 6.4 – Read section 6.4 in the textbook and complete associated exercises.**

**Trapezoidal Rule****Instruction – Learn about the trapezoidal rule.****Example Problem: Approximating the Area under a Curve by Using the Trapezoidal Rule. How do we use the trapezoidal rule to approximate the area under a curve?****Example Problem: Comparing the Trapezoidal Rule to Other Area Approximations. How does the trapezoidal rule compare with the LRAM, RRAM, and MRAM?**

**Reading Lesson 6.5: Trapezoidal Rule****Instruction – Reading Lesson****Practice Problems 6.5 – Read section 6.5 in the textbook and complete associated exercises.****Project: Analyzing Driving Data****Show that the area under a curve equates to the distance traveled.**

**Unit 6 AP Practice Questions****Unit 6 Free-Response Questions****Complete free-response questions for Unit 6.**

**Unit Test****Unit Test Answers**

**Unit 7: Mathematical Modeling Using Differential Equations**

**Introduction to Unit 7****Instruction – Unit 7 Introduction**

**Slope Fields****Instruction – Learn about slope fields.****Example Problem: Finding a General Solution. How do you determine the general solution to a differential equation?****Example Problem: Finding a Particular Solution. Given additional information, how do you find the one particular solution to a differential equation?****Example Problem: Constructing a Slope Field. How might the graph of a differential equation appear?****Example Problem: Graphing the Particular Solution on a Slope Field.How do you sketch the particular solution to a differential equation?**

**Reading Lesson 7.1: Slope Fields and Euler’s Method****Instruction – Reading Lesson****Practice Problems 7.1 – Read section 7.1 in the textbook and complete associated exercises.**

**Antidifferentiation by Substitution****Instruction – Learn about antidifferentiation by substitution.****Example Problem: Evaluating Indefinite Integrals. How do you evaluate indefinite integrals without changing variables?****Example Problem: Applying Indefinite Integrals to Velocity and Position. How do you apply indefinite integrals to functions of velocity and position?****Example Problem: Using Substitution to Evaluate Indefinite Integrals. How do you use u-substitution to evaluate indefinite integrals?****Example Problem: Evaluating Definite Integrals Using Substitution. How do you use u-substitution to evaluate definite integrals?**

**Reading Lesson 7.2: Antidifferentiation by Substitution****Instruction – Reading Lesson****Practice Problems 7.2 – Read section 7.2 in the textbook and complete associated exercises.**

**Exponential Growth and Decay****Instruction – Learn about exponential growth and decay.****Example Problem: Solving Differential Equations. How does solving a differential equation by separating the variables compare to solving previous differential equations?****Example Problem: Solving Exponential Growth Problems. How do you solve a problem involving exponential growth?****Example Problem: Solving Exponential Decay Problems. How do you solve a problem involving exponential decay?****Example Problem: Solving Cooling Problems. How can we model the change in temperature of an object as it cools?**

**Reading Lesson 7.4: Exponential Growth and Decay****Instruction – Reading Lesson****Practice Problems 7.4 – Read section 7.4 in the textbook and complete associated exercises.**

**Unit 7 AP Practice Questions****Unit 7 Free-Response Questions****Complete free-response questions for Unit 7.**

**Unit Test****Unit Test Answers**

**Unit 8: Applications of Definite Integrals**

**Introduction to Unit 8****Instruction – Unit 8 Introduction**

**Integral as Net Change****Instruction – Learn about integral as net change.****Example Problem: Calculating Displacement and Position. How do you calculate displacement and position given a particular velocity function?****Example Problem: Finding Total Distance Traveled. How do you calculate the total distance traveled given a particular velocity function?****Example Problem: Finding Net Change from Rate of Change Functions. How do you find the net change of a quantity from a rate of change function?****Example Problem: Finding Net Change from Graphs and Tables. How do you find the net change of a quantity from a rate of change given in graphical or tabular form?**

**Reading Lesson 8.1: Accumulation and Net Change****Instruction – Reading Lesson****Practice Problems 8.1 – Read section 8.1 in the textbook and complete associated exercises.**

**Areas in the Plane****Instruction – Learn about areas in the plane.****Example Problem: Calculating Area with dx Integration. How do you determine area by integrating with respect to x?****Example Problem: Calculating Area with dy Integration. How do you determine area by integrating with respect to y?****Example Problem: Calculating Area Using Subregions. How do you determine area by integrating using subregions?**

**Reading Lesson 8.2: Areas in the Plane****Instruction – Reading Lesson****Practice Problems 8.2 – Read section 8.2 in the textbook and complete associated exercises.**

**Volumes****Instruction – Learn about volumes.****Example Problem: Calculating Volumes with Disks. How do you determine volumes of solids using disks?****Example Problem: Calculating Volumes with Disks (dy). How do you determine volumes of solids by integrating with respect to y?****Example Problem: Calculating Volumes with Washers. How do you determine volumes of solids using washers?****Example Problem: Calculating Volumes with Known Cross Sections. How do you determine volumes of solids with known cross sections?**

**Reading Lesson 8.3: Volumes****Instruction – Reading Lesson****Practice Problems 8.3 – Read section 8.3 in the textbook and complete associated exercises.**

**Applications from Science and Statistics****Instruction – Learn about applications of science and statistics.****Example Problem: Calculating the Work Done in Stretching a Spring. How much work is done in stretching a spring?****Example Problem: Finding the Work Done When the Force Is Nonlinear. How do you calculate the work done when the force is nonlinear?****Example Problem: Finding Constant Fluid Force and Fluid Pressure. How do you calculate the fluid force and pressure when the pressure is constant?****Example Problem: Finding Nonconstant Fluid Force and Fluid Pressure. How do you calculate a nonconstant fluid force and pressure?****Example Problem: Using the Definite Integral to Find the Probability of an Event.How do you find the probability of an event using the probability density function?**

**Reading Lesson 8.5: Applications from Science and Statistics****Instruction – Reading Lesson****Practice Problems 8.5 – Read section 8.5 in the textbook and complete associated exercises.**

**L’Hospital’s Rule and Other Applications****Instruction – Learn about L’Hospital’s rule and other applications.****Example Problem: Applying L’Hospital’s Rule to Evaluate Limits. How do you apply L’Hospital’s rule to evaluate the limit of the indeterminate form 0/0?****Example Problem: Applying L’Hospital’s Rule to Evaluate Limits. How do you apply L’Hospital’s rule to evaluate the limit of the indeterminate form infinity/infinity?****Example Problem: Applying L’Hospital’s Rule to Other Indeterminate Forms. How do you apply L’Hospital’s rule to evaluate the limit of other indeterminate forms?****Example Problem: Comparing Growth Rates**

**Reading Lesson 9.2 and 9.3: L’Hospital’s Rule and Other Applications****Instruction – Reading Lesson****Practice Problems 9.2 and 9.3 – Read section 9.2 and 9.3 in the textbook and complete associated exercises.****Project: Calculating the Volume of an Object****Use calculus to calculate the volume of an object.**

**Unit 8 AP Practice Questions****Unit 8 Free-Response Questions****Complete free-response questions for Unit 8.**

**Unit Test****Unit Test Answers**

**Unit 9: Review**

**Preparing for the Exam****Warm-Up – Get ready for the lesson****Instruction – Preparing for the AP Calculus Exam**

**Review: Limits and Continuity****Instruction – Limits and Continuity**

**Review: Derivatives****Instruction – Derivatives**

**Review: Applications of Derivatives****Instruction – Applications of Derivatives**

**Review: Integrals****Instruction – Integrals**

**Review: Applications of Integrals****Instruction – Applications of Integrals**

**Review: Differential Equations****Instruction – Differential Equations**

**Practice Exam 1 – Part A****Instruction – AP Calculus Practice Exam 1 Instructions****Practice Exam Multiple Choice Section – Part A Answers****Practice Exam 1 – Part B****Practice Exam Multiple Choice Section – Part B Answers****Practice Exam 1 – Free Response Section****Instruction – The Free-Response Section****Free-Response Section – Complete Part A of the free-response section****Free-Response Section – Complete Part B of the free-response section.**

**Practice Exam 2 – Part A****Instruction – AP Calculus Practice Exam 2 Instructions****Practice Exam Multiple Choice Section – Part A Answers****Practice Exam 2 – Part B****Practice Exam Multiple Choice Section – Part B Answers****Practice Exam 2 – Free Response Section****Instruction – The Free-Response Section****Free-Response Section – Complete Part A of the free-response section.****Free-Response Section – Complete Part B of the free-response section**

**Unit 10: Cumulative Exam**

**Cumulative Exam****Cumulative Exam Answers**