### Reading Assignments - Math 101 Calculus I - Spring 2017

Be sure to check back, because this may change during the semester.

All numbers indicate sections from Calculus by Taalman and Kohn.

#### For Friday January 27 (Due 1/26 @ 8:00 pm)

Section 0.4 Exponential and Trigonometric Functions

• Topics:
• Exponential Functions
• Logarithmic Functions
• Examples: 1a, 2a, 3ab

1. How are the functions f(x)=2x and g(x)=log2(x) related?
2. Solve for x in the equation log2(x) + log2(x3)=12.

#### For Monday January 30 (Due 1/29 @ 8:00 pm)

Section 0.4 Exponential and Trigonometric Functions

• Topics:
• Trigonometric Functions
• Examples: 1b, 2b, 3c

1. What is 120 degrees equal to in radians?
2. What is the period of the sine function? How can you tell from the graph?

#### For Wednesday February 1 (Due 1/31 @ 8:00 pm)

Section 2.1 An Intuitive Introduction to Derivatives

• Topics:
• Slope Functions
• Position and Velocity
• Approximating the Slope of a Tangent Line
• Examples: 1, 2, 3

1. Look at the leftmost graph on page 156. Is the function f positive or negative at x=0? How about the derivative of f at x=0? Explain.
2. Let f(x)=x2. Find the slope of the secant line connecting the points (1,1) and (3,9) on the graph y=f(x).

#### For Friday February 3 (Due 2/2 @ 8:00 pm)

Section 1.1 An Intuitive Introduction to Limits

• Topics:
• Examples of Limits
• Limits of Functions
• Infinite Limits, Limits at Infinity, and Asymptotes
• Examples: 1, 2, 3

1. Why do you think we're studying limits now?
2. If f(x)=x2, what is limx→2f(x)? Explain.
3. If f(x)=1/x, what is limx→0f(x)? Explain.

#### For Monday February 6 (Due 2/5 @ 8:00 pm)

Section 1.4 Continuity and Its Consequences

• Topics:
• Defining Continuity with Limits
• Types of Discontinuities
• Continuity of Very Basic Functions
• Extreme and Intermediate Values of Continuous Functions
• Examples: 1, 3, 5

1. How can you tell from the graph of y=f(x) if the function f(x) is continuous?
2. Give an example of a function that is discontinuous at x = 3.
3. Give an intuitive explanation for why the Intermediate Value Theorem is true.

#### For Wednesday February 8 (Due 2/7 @ 8:00 pm)

Section 1.6 Infinite Limits and Indeterminate Forms

• Topics:
• Infinite Limits
• Limits at Infinity
• Indeterminate and Non-Indeterminate Forms
• Examples: 1, 2ab, 3

1. Give an example of a function that has a vertical asymptote at x = 3.
2. Give an example of a function that has a horizontal asymptote at y = -3.

#### For Friday February 10 (Due 2/9 @ 8:00 pm)

Section 2.2 Formal Definition of the Derivative

• Topics:
• The Derivative at a Point
• The Derivative as a Function
• Differntiability
• Tangent Lines at Local Linearity
• Examples: 1 - 4

1. If f(x) is a function, give two different interpretations of the value f '(a), the derivative of f at x = a. (Hint: Think about Section 2.1)
2. What is an advantage of determining f '(x), the derivative function, rather than f '(a), the derivative at a specific point x=a?

#### For Monday February 13 (Due 2/12 @ 8:00 pm)

Section 2.3 Rules for Calculating Basic Derivatives

• Topics:
• Derivatives of Linear Functions
• The Power Rule
• The Constant Multiple and Sum Rules
• The Product and Quotient Rules
• Examples: 1, 2, 4

1. If f(x)=x8, what is f '(x)?
2. If f(x)=x1/3 (the cube root of x), use the graph of y=f(x) to explain why f '(0) does not exist.
3. Explain what is wrong with the following calculation and fix it:
f(x)= (x2 + 7x) (x3 - 5x +9)
f '(x)=(2x+7)(3x2-5)

#### For Wednesday February 15

Re-read Section 2.3 Rules for Calculating Basic Derivatives

#### For Friday February 17 (Due 2/16 @ 8:00 pm)

Section 2.4 The Chain Rule and Implicit Differentiation

• Topics:
• Differentiating Compositions of Functions
• Examples: 1, 2

Explain what is wrong with the following calculations and fix them.

1. f(x)= (x2+2x)130
f '(x)=130(x2+2x)129

2. f(x)= (x3+8x^2)12
f '(x)=12(3x2+16x)11

#### For Monday February 20

Q&A for Exam 1. No Reading Assignment.

#### For Wednesday February 22

Section 2.5 Derivatives of Exponential and Logarithmic Functions

• Topics:
• Derivatives of Exponential Functions
• Exponential Functions Grow Proportionally to Themselves
• Derivatives of Logarithmic Functions
• Examples: 1, 3

1. What is the 42nd derivative of f(x)=ex?
2. If g(x)=x ln(x) - x, find g'(x).

#### For Friday February 24 (Due 2/23 @ 8:00 pm)

Section 2.6 Derivatives of Trigonometric and Hyperbolic Functions

• Topics:
• Derivatives of Trigonometric Functions
• Examples: 1

1. What is the value of limθ -> 0 sin(θ) / θ ? (Hint: Look at the graph)
2. What is the value of limθ -> 0 (1-cos(θ)) / θ ? (Hint: Look at the graph)
3. Why do we care about these limits?

#### For Monday February 27 (Due 2/26 @ 8:00 pm)

Re-read Section 2.6 Derivatives of Trigonometric and Hyperbolic Functions

#### For Wednesday March 1 (Due 2/28 @ 8:00 pm)

Section 3.1 The Mean Value Theorem

• Topics:
• The Derivative at a Local Extremum
• The Mean Value Theorem
• Examples: 1, 4, 5

1. If f(x)= 3x2 - 6x, is x=1 a critical value? Why or why not? How about x=0?
2. Explain the Mean Value Theorem using "car talk" (that is, in terms of velocity).

#### For Monday March 6 (Due 3/5 @ 8:00 pm)

Section 3.2 The First Derivative and Curve Sketching

• Topics:
• Derivatives and Increasing/Decreasing Functions
• Functions with the Same Derivative
• The First Derivative Test
• Examples: 1, 2, 3

Let f(x) = x4 - 4 x3 + 2

1. Show that f has critical points at x=0 and x=3.
2. Use the First Derivative Test to classify the critical point at x=3 as a local maximum, a local minimum, or neither.
3. Use the First Derivative Test to classify the critical point at x=0 as a local maximum, a local minimum, or neither.

#### For Wednesday March 8 (Due 2/28 @ 8:00 pm)

Section 3.3 The Second Derivative and Curve Sketching

• Topics:
• Derivatives and Concavity
• Inflection Points
• The Second-Derivative Test
• Curve-Sketching Strategies
• Examples: 1, 2, 3

Let f(x) = x4 - 6 x2 + 8x + 2

1. Show that f has critical points at x=-2 and x=1.
2. Use the Second Derivative Test to classify the critical point at x=-2 as a local maximum, a local minimum, or neither.
3. Use the First Derivative Test to classify the critical point at x=1 as a local maximum, a local minimum, or neither.

#### March 13 - 17

Spring Break. Surprisingly, no Reading Assignments.

#### For Monday March 20 (Due 3/19 @ 8:00 pm)

Section 3.4 Optimization

• Topics:
• Finding Global Extrema
• Translating Word Problems into Mathematical Problems
• Examples: 1, 2

Let f(x) = x4 - 6 x2 + 8x + 2

1. On the interval [-3,2], at which x-values can the global maximum and global minimum of f(x) occur?
2. What is the global minimum of f(x) on the interval [-3,2]? What is its global maximum on this interval?

#### For Wednesday March 22

Re-read Section 3.4 Optimization, including Examples 3 & 4.

#### For Friday March 24

Re-read Section 3.4 Optimization, including Examples 3 & 4.

#### For Monday March 27

Q&A for Exam 2. No Reading Assignment.

#### For Monday April 3 (Due 4/2 @ 8:00 pm)

• Topics:
• Accumulation Functions
• Sigma Notation
• The Algebra of Sums in Sigma Notation
• Examples: 1, 2, 3

1. Calculate Σ5k=1   3k2
2. Why do you think we are introducing sigma notation now?

#### For Wednesday April 5 (Due 4/4 @ 8:00 pm)

Section 4.2 Riemann Sums

• Topics:
• Subdivide, Approximate, and Add Up
• Approximating Area with Rectangles
• Riemann Sums
• Types of Riemann Sums
• Examples: 1, 2, 4

1. What is the purpose of a Riemann sum?
2. If f(x)=x2, will a right sum for f on [0,2] underestimate or overestimate the area under the curve? Why?

#### For Monday April 10 (Due 4/9 @ 8:00 pm)

Section 4.3 Definite Integrals

• Topics:
• Defining the Area Under a Curve
• Properties of Definite Integrals
• Examples: 1, 4, 5

1. Find ∫01 2x dx
2. Will the definite integral ∫02 x-5 dx be positive or negative? Why?

#### For Wednesday April 12 (Due 4/11 @ 8:00 pm)

Section 4.4 Indefinite Integrals

• Topics:
• Antiderivatives and Indefinite Integrals
• Antidifferentiation Formulas
• Antidifferentiating Combinations of Functions
• Examples: 1, 2, 3, 4

1. What is the difference between a definite integral and an indefinite integral?
2. Find ∫ 2x + cos(x) dx

#### For Monday April 17 (Due 4/16 @ 8:00 pm)

Section 4.5 The Fundamental Theorem of Calculus

• Topics:
• The Fundamental Theorem
• Using the Fundamental Theorem of Calculus
• Examples:1, 2, 3

1. Find the area of the region above the x-axis and below the graph of f(x)= 4/x + cos(x) + 1 between x=1 and x=10.
2. Does it matter which antiderivative F you pick to use in the Fundamental Theorem? Explain.

#### For Wednesday April 19 (Due 4/18 @ 8:00 pm)

Section 4.7 Functions Defined by Integrals

• Topics:
• Area Accumulation Functions
• The Second Fundamental Theorem of Calculus
• Examples:1, 2, 3

Does every continuous function have an antiderivative? Why or why not?

#### For Monday April 24

Section 5.1 Integration by Substitution

• Topics:
• The Art of Integration
• Undoing the Chain Rule
• Choosing a Useful Substitution
• Finding Definite Integrals by Using Substitution
• Examples: 1, 2, 3, 5

1. Substitution attempts to undo one of the techniques of differentiation. Which one is it?
2. Use u-substitution to find an antiderivative of f(x) = 3x2 cos(x3)
3. Explain why ∫ cos(x) sin(x)2 dx and ∫ ln(x)2 / x dx are essentially the same integral after performing a substitution.

#### For Monday May 1

Q&A for Exam 3. No Reading Assignment.

#### For Wednesday May 3

The BIG Picture. No Reading Assignment.

#### For Friday May 5

The BIG Picture. No Reading Assignment.

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