Reading questions:

1. Show that y(x)=x^(1/3) is a solution to the differential equation y'(x)=1/(3*y^2).

   SS says (but I re-ordered, to make it a bit more readable on the web)
   

   y'(x)	=	(1/3)x^(-2/3)
   	=	1/(3*x^(2/3))
   	=	1/(3*(x^(1/3))^2)
   	=	1/(3*y^2)

   Since y(x)=x^(1/3) makes the DE true, it is one solution to the DE.

2. Solve the initial value problem y'(x)=5/x^2+4, y(1)=12.

   SS also says:
   

   if y'(x)=5/x^2+4 then y(x)=(-5/x)+4x+C, so y(1)=-5+4+C =-1+C

   Since y(1) has to equal 12 then C must equal 13...

   so y(x)=(-5/x)+4x+13

Due Friday 4/18 at 9am

Section 5.6 Approximating Sums: The Integral as a Limit

Reading questions:

1. Explain, in your own words, the idea of using Riemann Sums to approximate integrals.

   NT says:
   

   A Riemann sum allows you to closely approximate the signed area. It finds the area of a series of rectangles that approximate the shape of the f(x).

2. If f(x) is decreasing on [a,b], will L_n underestimate or overestimate the integral of f from a to b? How about R_n?

   GC says:
   

   Ln will over estimate and Rn will underestimate.

Due Monday 4/14 at 9am

Section 5.4 Finding Antiderivatives: The Method of Substitution

Reading questions:

1. Explain the fundamental difference between a definite integral and an indefinite integral. Please go deeper than saying one has limits of integration and one doesn't. The first is a real number -- why? What does it represent? Then think similarly about an indefinite integral? Is it a real number? If not what is it? Why? What does it represent?

   MS and AK say:
   

   A definite integral looks to find the signed area of a function over a specific interval.

   Indefinite integrals are used to denote the ÒfamilyÓ of all possible antiderivatives of f.

2. Substitution attempts to undo one of the techniques of differentiation. Which one is it?

   AJK says:
   

   The chain rule

3. What are the three steps in the process of substitution?

   DH says:
   

   The first step is to choose a function u=u(x) and come up with a function du=u'(x)dx. Then substitute u and du in to the original integral to produce a new integral.

   The second step is to antidifferentiate in u.

   Then resubstitute --substitute back to eliminate u.

Due Wednesday 4/9 at 9am

Section 5.3 The Fundamental Theorem of Calculus

Reading questions:

1. Find the area between the x-axis and the graph of f(x)=x^3+4 from x=0 to x=3.

   JS says:
   

   First, I would find the anti-derivative, which is: x^4/4+4x.

   Then, since we want to find the area from 0 to 3, which is calculated using F(b) - F(a), this area turns out to be 32.25.

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

   NM says:
   

   Every continuous function has an anti-derivative, since A_f must always be an anti-derivative of f.

3. If f(x)=3*x-5 and a=2, where is A_f increasing? Decreasing? Why?

   NW says:
   

   The area between the function and the x axis is the antiderivative of the function, so area is increasing when the original function is positive, and decreasing when it is negative.

   

   It is decreasing at all values less than 5/3 and increasing on all values larger then that.

4. How would your answer change if a=0?

   BF says:
   

   It doesn't.

Due Monday 4/7 at 9am

Section 5.2 The Area Function

Reading questions:

1. Let f be any function. What does the area function A_f(x) measure?

   AK says:

   It measures the signed area defined by f (and the horizontal axis) from a to x.

2. Let f(t)=t and let a=0. What is A_f(1)?

   NP says:
   

   A_f(1) = 1/2 since 1/2*1*1= 1/2

Due Friday 4/4 at 9am

Section 5.1 Areas and Integrals

Reading questions:

1. What does the integral of a function f from x=a to x=b measure?

   DD says:
   

   The integral of a function f from x=a to x=b measures signed area. The integral of f from x=a to x=b implies a direction : x starts at a and ends at b.

2. Is the integral of f(x)=5x from x=-1 to x=3 positive or negative?

   I say:
   

   The integral of 5x from -1 to 3 is the signed area of the region between the line y=5x and the x-axis, from x=-1 to x=3. This signed area is shown to the left. Area below the x-axis is negative, area above the x-axis is positive. More of the area is above the x-axis than is below, and so the total area turns out to be positive.

Due Wednesday 4/2 at 9am

Section 4.9 Why Differentiability: The Mean Value Theorem

Reading questions:

1. What are the hypotheses of the Mean Value Theorem?

   KG says:
   

   The hypotheses of the Man Value Theorm state that the function y=f(x) must be continous on the closed internal and differential on the open interval.

2. What is the conclusion of the Mean Value Theorem?

   AB says:
   

   (f(b)-f(a))/(b-a)gives the slope of the line joining the endpoints of the curve. for an x between [a,b], the derivative must equal the slope of the line joining the endpoints.

3. Explain the MVT using "car talk" -- that is, using velocity.

   SP says:
   

   If you are averaging 60 mph in a car over a given period of time, you must have been going exactly 60 mph at an instant within that time period.

Due Friday 3/28 at 9am

Section 4.8 Why Continuity Matters

Reading questions:

1. What are the hypotheses of the Intermediate Value Theorem?

   NP says:
   

   The hypotheses of the Intermediate Value Theorem: f is continuous on the closed, bounded interval [a,b], and y is any number between f(a) and f(b).

2. What is the conclusion of the Intermediate Value Theorem?

   CM says:
   

   The conclusion of the Intermediate Value Theorem is that for some input c between a and b, f(c)=y.

   In other words, as JL says:
   

   On the way from (a,f(a)) to (b,f(b)), the graph of f crosses every horizontal line between y= f(a) and y= f(b).

Due Wednesday 3/19 at 9am

Section 4.7 Building Polynomials to order: Taylor Polynomials

Reading questions:

1. Why would you want to find the Taylor polynomial of a function?

   AL says:
   

   We find a Taylor polynomial of a function because Taylor polynomials approximate other (non-polynomials) functions.

   A Taylor polynomial would be useful if a function is ill-behaved, complicated, or poorly understood. Thus you replace it with a Taylor polynomial g(x). By replacing a function with a Taylor polynomial you can approximate the difficult equation. This allows you go get a general idea/understanding of the function.

2. In your own words, briefly explain the idea of building the Taylor polynomial for a function f(x).

   SK says:
   

   To build a Taylor Polynomial of order n at x0, you first need to find n derivatives of f(x). Then, you take the sum illustrated in the equation below:

   Pn(x)=f(x0) + f'(x0)(x-x0) + f''(x0)/2! (x-x0)^2... etc.

Due Monday 3/17 at 9am

Section 4.3 Optimization

Reading questions:

1. At which x-values can a continuous function f(x) achieve its maximum or minimum value on a closed interval [a,b]?

   AB says:
   

   On a closed interval [a,b], a continuous function can achieve its maximum or minimum value only at critical points in (a,b) or at endpoints of [a,b].

2. What is the difference between an objective function and a constraint equation?

   MS says:
   

   Objective functions describe the quantities to be maximized or minimized
   A constraint function describes a condition that must be satisfied by the variables in an optimization problem.

Due Monday 3/3 at 9am

Section 3.2 Composition and the Chain Rule

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

1. f(x)=sin(x^2). f ' (x) = cos(x^2)+sin(2*x)

   NT says:
   

   In the example, the product rule was erroneously used instead of the chain rule.

   f'(x)= cos(x^2)*2x.

2. g(x)=exp(3*x). g ' (x)=exp(3*x).

   BF says:
   

   The chain rule was not applied to find the derivative of the function. You forgot to multiply by the derivative of 3x.

   So, g'(x)= 3*exp(3*x).

3. h(x)=(sin(x))^3. h ' (x)= 3*(cos(x))^2.

   KP says:
   

   According to the chain rule you have to take the derivative of the outside function composed with the inside function and then multiply it by the derivative of the inside function.

   THe correction would be h'(x)= 3*(sin(x))^2*cos(x).

Due Friday 2/29 at 9am

Section 3.1 Algebraic Combinations: The Product and Quotient Rules

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

1. f(x)=x^2*sin(x). f ' (x)=2*x*cos(x).

   MTS says:
   

   According to the product rule, the calculation should look like:
   2x*sin(x)+x^2*cos(x)

2. g(x)=sin(x)/(x^2+1). g ' (x) = cos(x)/(2*x).

   SS says:
   

   In the derivative above, you took the derivative of the first part and divided it by the derivative of the second part. You were supposed to use the quotient rule.

   The derivative below is correct:
   g'(x) = (x^2+1)(cos(x))-(sin(x))(2x) / (x^2+1)^2

-->

Due Wednesday 2/27 at 9am

Section 2.7 Derivatives of Trigonometric Functions: Modeling Oscillation

Reading questions:

1. What is limit( (cos(h)-1)/h, h=0)?

   Combining two answers, ACC and NM say:
   

   Since cos(0)=1, we can re-write the expression as
   limit( (cos(h)-cos(0))/(h-0), h=0). This defines the derivative of the cosine function at x=0.

   Because the cosine graph is horizontal at x=0, the limit of ( (cos(h)-1)/h, h=0) is zero.

2. What is limit( sin(h)/h, h=0)?

   EP says:
   

   The limit( sin(h)/h, h=0) defines the derivative of the sine function at x=0. After looking at the slope of the curve y=sin x near x=0, we can conclude that the derivative is 1.

3. Why do we care about the limits in the first two questions?

   MPS says:
   

   We care about these two limits because they help us define the derivatives of the sin and cos functions.

4. Differentiate sin(2x).

   VB says:
   

   (sin(2x))' =2cos(2x)

Due Monday 2/25 at 9am

Section 2.6 Derivatives of Exponential and Logarithmic Functions; Modelling Growth

Reading questions:

1. What is the 47th derivative of f(x)=exp(x)?
   exp(x) is Maple notation for the function e^x.

   KG says:
   

   The 47th derivative of f(x)= exp(x) is exp(x), it doesn't change.

2. Do exponential functions model population growth well? Explain.

   AK says:
   

   A compound-interest bearing account exhibits exponential growth since it grows at a rate proportional to its size.

Due Friday 2/22 at 9am

Section 2.4 Using Derivative and Antiderivative Formulas

Reading questions:

1. Explain in your own words what an antiderivative of a function g(x) is.

   AK says:
   

   A function F is an antiderivative of f if FÕ(x) = f(x). The process of recovering a function F(x) from its derivative f(x) is called antidifferentiation.

   DH adds:

   To find the antiderivative, you do the complete opposite as when finding the derivative.

   For example, the antiderivative of 5x^27+32x^5+10 is 5x^28/28+32x^6/6+10x. Instead of subtracting (from the power),you add, and instead of multiplying by the power, you divide. You must remember to take care of the power first before doing the dividing.

2. How many antiderivatives does f(x)=3x^2 have? Why?

   SK says:
   

   f(x)=3x^2 has an infinite number of antiderivatives because adding a constant to any function F has no effect on F'.

Due Wednesday 2/20 at 9am

Section 2.3 Limits

Reading questions:

1. Let g(x)=(x^2-9)/(x-3), as in Example 2.
   1. Is g(x) defined at x=3? Why or why not?

      AMC says:
      

      g(x) is not defined at x=3 because that would mean the denominator would be zero and you can't divide anything by zero.

   2. What is limit(g(x),x=3)? Why?

      CM says:
      

      The limit at x=3 is 6 because the equation will equal a number close to 6 for any number close to x=3.

      AS adds:
      

      Lim _x->3g(x)= Lim_x->3 (x+3) =6

2. Is f(x)=|x| continuous at x=0? Why or why not?

   DD says:
   

   F(x)=|x| is continuous because even though there is a kink it doesnÕt mean that the graph is broken.

Due Friday 2/15 at 9am

Section 2.2 Derivatives of Power Functions and Polynomials

Reading questions:

1. What is the derivative of f(x)=x^3?

   PE says:
   

   f '(x)=3x^2

2. Let f(x)=x^(1/3) (i.e. the cube root of x). Use the graph of y=f(x) to explain why f'(x) does not exist at x=0.

   ACC says:
   

   It does not exist at x=0 because the graph has a vertical tangent line.

   Graph of x^(1/3)

Due Monday 2/11 at 9am

Section 2.1 Defining the Derivative

Reading questions:

1. Let f(x)=x^3. Find the slope of the secant line from x=-2 to x=4.

   AJL says:
   

   Use the difference quotient.

   If f(x)=x^3, (f(4)-f(-2))/6=12.

2. For a function f, what does the difference quotient ( f(a+h) - f(a) )/h measure?

   ASL says:
   

   ( f(a+h) - f(a) )/h measures the slope of the secant line that is created by two points on any given graph.

   
   AMC adds:
   

   The difference quotient measures the average rate of change of the function f from x=a to x=a+h.

3. Let f(x)=x^3. What is the average rate of change of f from x=-2 to x=4?

   VB says:
   

   To find the average rate of change of f from x=-2 to x=4, I use the same method as question one. f(4)-f(-2)/6= 12. The average rate of change is found by using the difference quotient.

Due Friday 2/8 at 9am

Section 1.7 The Geometry of Higher Order Derivatives

Reading questions:
Use the graphs of f, f ', f '' in Figure 3 on page 67.

1. By looking at the graph of f '', how can you tell where f is concave up and concave down?

   AG says:
   

   When f'' is positive, f' is increasing which says that the function f is concave up. When f'' is negative, f' is decreasing which means the function f is concave down.

2. By looking at the graph of f ', how can you tell where f is concave up and concave down?

   KP says:
   

   The graph of f will be concave up when f' is increasing, and f will be concave down when f' is decreasing.

   At point B on graph of f', the graph is changing direction from decreasing to increasing, and at point B on the graph of f, f is changing concavity.

Due Wednesday 2/6 at 9am

Section 1.6 The Geometry of Derivatives

Reading questions:
Look at the graph of f ' in Example 2.

1. Where does f have stationary points? Why?

   NT says:
   

   f has stationary points at x= -1, x=0, and x=1 because on the graph of f ' the graph crosses the x axis. This means that the slope of f at these points is 0, making them stationary points (of f).

2. Where is f increasing? Why?

   AB says:
   

   f is increasing from [-1.5,-1) and from (1, 1.5) because the derivative is positive, therefore the slope of f is positive, so f is increasing in those intervals.

3. Where is f concave up? Why?

   GC says:
   

   f is concave up from -0.5 to 0 and from 0.5 to infinity because these are the points where the f '(x) graph slopes upward.

Due Monday 2/4 at 9am

Section 1.5 Estimating the Derivative

Reading questions:

1. What does the term locally linear mean?

   JS says:
   

   The term locally linear means that by zooming in on any graph will expose a straight line, or a near straight line. Even if the graph appears curvy from a distance, by zooming in, that small piece of the graph will become straight enough to discern its slope and therefore its derivative.

2. Explain why the derivative of f(x)=|x| does not exist at x=0.

   EP says:
   

   The derivative of f(x)=|x| does not exist at x=0 because the f(x)=|x| function has a kinked graph, which cannot be straightened out by zooming. If zooming has no effect at all on the kink at the origin, then, in mathematical terms f'(0) does not exist.

Due Friday 2/1 at 9am

Section 1.4 Amount Functions and Rate Functions: The Idea of the Derivative

Reading questions:
Look at the graphs of P(t) and V(t) in Figure 1 on page 37.

1. Is the derivative of P positive or negative at t=5? Explain.

   EB says:
   

   The derivative of P at t=5 is negative because v(t) is equal to the derivative of p(t) and on the graph v(t)=-30 when t=5.

2. Is the second derivative of P positive or negative at t=5? Explain.

   BF says:
   

   the second derivative of P is represented by A(t) -- also known as the derivative of V(t). at t=5 on the V(t) graph, the velocity is increasing, so A(t) would be positive.

3. give a value of t where the derivative of P is zero.

   JL says:
   

   t=3, t=6.75. These are both points where velocity crosses the x-axis, meaning she is changing from traveling eastward to westward.

Due Monday 1/28 at 9am

Problem Set Guidelines
Section 1.3 A Field Guide to Elementary Functions

Reading questions:

1. How are the functions f(x)=3^x and g(x)=log[3](x) related?
   log[3](x) is Maple notation for log₃(x).

   SS says:
   

   The functions f(x)=3^x and g(x)=log[3](x) are related because they are inverses of each other. Inverses of each other means that they reflect across the line y=x.

2. What are some of the properties of sin(x)?

   PY says:
   

   Sin(x) has the following properties: Its domain is negative infinity to positive infinity, and its range is [-1,1]. The function Sin(x) is considered odd and repeats itself on intervals of 2pi.

Due Friday 1/25 at 9am

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Section 1.1: Functions, Calculus Style
Section 1.2: Graphs
Reading questions:

Let f(x)=x^2.

1. How is the graph of y=f(x)+3=x^2+3 related to the graph of y=f(x)? Why?

   NW says:
   

   The graph y=f(x)+3=x^2+3 is just the graph of f(x)=x^2 shifted up three units on the y-axis, with an x-intercept at (0,3). Adding a constant "a" to a function f(x) results in a vertical translation g(x)=f(x)+a

2. How is the graph of y=f(x+3)=(x+3)^2 related to the graph of y=f(x)? Why?

   NP says:
   

   The graph y = f(x + 3) = (x + 3)^2 is related to the graph of y = f(x) in the sense that the graph y = f(x + 3) = (x + 3)^2 is a leftward shift of the f(x) graph. This type of shift is known as a horizontal translation, where the graph y = f(x + a) is the result of shifting the f-graph a units to the left.

3. Which of f(x), f(x)+2, f(x+2) are even? odd?

   SP says:
   

   f(x) and f(x)+2 are both even because when you replace (x) with (-x) the equation will still equal f(x) (or f(x)+2). f(x+2)is neither even or odd because when you replace (x) with (-x) and square it you come up with different results:
   (-x+2)^2=x^2-4x+4,
   while (x+2)^2=x^2+4x+4.

Janice Sklensky
Wheaton College
Department of Mathematics and Computer Science
Science Center, Room 109
Norton, Massachusetts 02766-0930
TEL (508) 286-3973
FAX (508) 285-8278
jsklensk@wheatonma.edu