# Math 113 Projects

Spring 1999

CHAPTER 4

Project 4.7

The part of the graph of 3xy3 - x3y +2 = 0 that is near (2,1) defines y as a function of x implicitly. Let us refer to this function as y = f(x). We know that f(2) = 1.

1. Find [dy/ dx] by implicit differentiation.

2. Is f(x) increasing or decreasing near x = 2? Explain how you know.

3. Find the equation of the tangent line to the graph of 3xy3 - x3y +2 = 0 at (2,1).

4. Complete the table of estimates of values of y = f(x) for x near 2 using the method described:

 x
 1.96
 1.98
 2
 2.02
 2.04
 y

For each value of x, substitute in the equation and then use a calculator to solve for y. Illustration: To approximate y = f(x) for x = 1.99 substitue 1.99 for x in the equation to obtain 3(1.99)y3 - (1.99)3y + 2 = 0. Solve this equation on your calculator. You can use the SOLVE operation or you can graph the function 3(1.99)x3 - (1.99)3x + 2 . Then use the ZOOM and TRACE features to find the where the function is zero.

5. Find all points where the tangent line to the graph of 3xy3 - x3y +2 = 0 is horizontal. Explain your reasoning at each stage.

Project 4.8

1. (a) Find the best linear approximation to f(x) = lnx at x = 3.

(b) Use your calculator to plot f(x) and your linear approximation on the same set of axes. Sketch your plots below and explain how this confirms that your linear approximation is reasonable.

(c) Use your linear approximation to estimate f(2.9 ). Indicate the relevant points on your graph and determine if your estimate is too large or too small. Explain.

2. Find the sign of limx ® a[f(x)/ g(x)] from the figures for Exercises 7 and 8 on page 232 of the text. Explain.

Project E

Refer to Appendices A and E on finding roots using bisection and Newton's method.

1. What is the smallest positive zero of the function f(x) = sinx? Apply Newton's method, with intial guess x0 = 3, to see how fast it converges to pi = 3.1415926536.

(a) Compute the first two approximations, x1 and x2; compare x2 with pi. >. (b) Newton's method work's very well here. Explain why. (To do this, you will have to outline the basic idea behind Newton's method.) (c) Estimate the location of the zero using besiection, starting with the interval [3,4]. How does bisection compare to Newton's in terms of accuracy?

2. Newton's method can be very sensitive to your initial estimate, x0. For example, consider finding a zero of f(x) = sin(x) - 2/3x.

(a) Use Newton's method with the following initial estimates to find a zero:

x0 = 0.904, x0 = 0.905, x0 = 0.906,

(b) What happens?

CHAPTER 3

Project 3.1

Let f(x)=x2 on the interval [0,1]. Use the program RECT and sketch the graphs of the left- and right-hand sums for n=5,10,47. How does the area under the left- and right-hand sums compare to the area under the curve? Explain what you are seeing as n increases.

Fill in the following table rounding to 5 decimal places. (For n>47, use the program SUM.) The program requires several minutes for the larger values of n.

 n Left-hand right-hand Difference of sum sum sums 5 0.24 0.44 0.20000 10 0.285 0.385 0.10000 47 94 200 1000 2000 4000

What value is the left-hand sums heading for as n approaches infinity?

What value is the right-hand sums heading for as n approaches infinity?

What do you think is the area under the curve?

Project 3.2

We can find ò13(6-2x)  dx by realizing that this integral is the area under the straight line y=6-2x on the interval [1,3]. The area enclosed is a triangle with base equal to 2 and height equal to 4. (ò denotes the integral sign). Then

area = 1/2 (base) x (height) = 1/2(2)(4) = 4

Verify this by filling in the table below using the function f(x)=6-2x on the interval [1,3]. Use the program SUM.

 n Left-hand right-hand Difference of sum sum sums 10 50 100 200 1000 2000

Project 3.3

The integral ò02Ö{4-x2}  dx is one fourth of the area of a circle of radius 2 and thus is p. Notice that the function y = Ö{4-x2} is decreasing on the interval [0,2]. Thus the left-hand sums should be greater than the area p and the right-hand sums should be less than p.

(a) Draw a graph of the rectangles associated with both the left- and right-hand sums when n = 4.

(b) Fill in the table below and verify this.

(c) Also use the program MSUM that uses the rectangles with heights equal to the values of f at the midpoint of the subintervals. Compare the error in using MSUM with the errors in using the left- and right-hand sums.

 n Left-hand right-hand Difference of MSUM sum sum sums 10 50 100 200 1000 2000

Project 3.4

1. Given in Figure 3.44 on page 173 is the graph of the derivative f'(x). Sketch the graph of f(x), starting with f(0) = 2, and explain how you obtained it.

2. On the interval [0,p/2 ], the function f(x) = sin(x) lies under its tangent line at x = 0 and above the line through the points (0,0) and ( p/2,1).

(a) Make a sketch and use the information that the graph of f is between two lines to estimate the area under sin(x) and between a = 0 and b = p/2.

(b) Use right/left hand sums to find an approximation of the area up to an error of 1%.

Project Focus

1. Consider the definite integral ò02e-x2/4   dx. Determine without using a calculator, the most likely range for the value of this integral, and provide evidence for your answer.
(a) Is it less than -0.1?
(b) Is it more than 3?
(c) Is it between 2/e = 0.7357 and 2?

2. Aproximate the following two definite integrals. If posible compute the exact value. Explain.
(a) ò133x2   dx
(b) ò01x sin(1/x)   dx

CHAPTER 5

Project 5.1

1. Let f''(x) be a continuous function with one zero point.
(a) What is the maximal number of zeros that f(x) can have?
(b) What is the minimal number of zeros that f(x) can have?

2. Let f(x) be a continuous function with two inflection points and no local extrema. sketch a function f(x) and f'(x) if you also know that f'(0)=1.

3. Let f(x) be a positive differentiable function and let f(k) = 0 for all integers k except 0 and f(0) = 1. Sketch a possible function f(x) and its derivative.

Project 5.2

1. Let a be a positive real number and let fa,k(t) be a two-parameter family defined by y = a(1+tekt)
(a) Find all the critical points and classify them, ie determine are they local minimum, local maxima, inflection points or neither.
(b) Discuss what happens to the two-parameter family when you change a and k.
(c) Sketch the graph which will explain the relation between k and the extremal value of fa,k(t), i.e. for a fixed a draw a graph of the function g(k) where g(k) = extremal value of fa,k(t), i.e. the value attained at a local maximum or minimum.

2. The two-parameter family y = a(10 + tekt) models populaltion growth on some isolated island in the Pacific. The variable t represents time measured in years and t=0 corresponds to the year 1750 when the first census is recorded. The records of the first census show a population of 100 people. the island's community developed very well until 1800, when the outbreak of a deadly virus massacred the population of the island.
(a) find the parameters a and k so that fa,k(t) describes the growth of the population on the island.
(b) Find when the first people inhabited the island.
(c) What was the maximal number of people who lived on the island?
(d) What will happen with the population on the island in the future?

Project 5.3-5.4

Let q be the quantity of items. Let C(q)=10(6-e{-q2}) and R(q)=55 + 5 cos(q).

1. (a) Find bounds for C(q) and R(q) where x ³ 0.

(b) Graph C(q) and R(q) on the same graph.

2. (a) Find functions for Marginal Cost and Marginal Revenue.

(b) On your calculator, set your X-window to be 0 to 27; set your Y-window to be -5 to 5. Sketch the graphs for Marginal Cost and Marginal Revenue on the same set of axes. (Marginal Cost should go very close to zero fairly quickly.)

(c) Using the TRACE function on your calculator, find the first eight values for q where the profit function may have a local extrema.

(d) From these eight values, determine which values correspond to local maxima and which values correspond to local minima for the profit function..

(e) Can you use your info in part (d) to determine where the profit function has a global maximum? If so, determine the value of q where profit has a global maximum.

Project 5.5

On the same side of a straight river are two towns; there is a pumping station to be built which will service the two towns. The pumping station (S) is to be built at the river's edge with the pipes extending straight to the towns. The location of the towns are shown in Figure 5.71 on page 276. Answer the following questions which deal with minimizing the total length of pipe.

1. Find an expression for l1 in terms of x.

2. Find an expression for l2 in terms of x.

3. For the following values of x, fill in the chart finding the total length of pipe for each x.

 x (mi.) l1 (mi.) l2 (mi.) L = total length of pipe (mi.) 0 . . . 1 . . . 2 . . . 3 . . . 4 . . .

4. According to the chart above, what would be a good guess for the position of the pumping station?

5. Now, to find the exact value for x, find an expression for L in terms of x and minimize it. NOTE: There is quite a bit of algebra involved so be careful and check your steps.

Project 5.6

1. Prove the following idendity: sinh(x+y) = sinh(x)cosh(y) + cosh(x)sinh(y)

2. Rewrite the following expression in terms of exponentials, simplifying as much as possible. 1/[cosh(2x) - sinh(2x)].

3. Show that if y = sin-1(tanh(x)) then y' = 1/[cosh(x))].

CHAPTER 6

Project 6.1

1. Given the function f shown in Problem 6 on page 297, assume F(x) is such that F'(x) = f(x).
a) What are the critical points of F(x)?
(b) Which critical points are local maxima, local minima, neither?
(c) Sketch a possible graph of F(x).

2. Assume f'(x) is given by the graph in Figure 6.10 on page 297.
a) Sketch a graph of f.
(b) Find f(0) and f(7).
(c) Find ò07f'(x)  dx = 21 in two different ways.
(c)

Project 6.2

1. Using the Fundamental Theorem of Calculus, find the limits of integration (i.e. find c and c+3) for òcc+32x  dx = 21

2. The average value of the function f(x) = 1 - 2x on the interval [0,q] is equal to -2. Find q.

3. Sketch the region bounded by the given curves andfind the area of the region.
y = x2, y = 2x + 5, x = 0, x = 6

Project 6.3

When an object is thrown vertically upward from ground level with an initial velocity of 64 ft/sec:
a. Find a formula for its height above the ground after seconds.
b. When does it reach its highest point?
c. How high is that point?
d. How long does it take to fall back to the ground?
e. With what velocity does it strike the ground?

Project 6.4

1. Make a table of values for the function Si(x) = ò0xsin(t)/t  dt for x = 0,1,2,3,4,5.

Since the integrand is undefined at t = 0, take the lower limit as 0.0001 instead of 0. Where is Si(x) increasing? Where is Si(x) decreasing? Explain your answer.

2. Let F(x) = ò0xsin(2t)  dt

(a) Evaluate F(p)
(b) Draw a sketch to explain geometrically why the answer is correct.
(c) For what values of x is F(x) positive? negative?

3. Find the derivative d/dx[ò1x(1+t)200  dt

CHAPTER 7

Project 7.1

Evaluate the following integrals.

1. ò(3x-1)/(3x2 -2x +1)   dx

2. ò sin3(x) cos(x)   dx

3. ò1/(2x-1)   dx

Project 7.2

1. Find ò1/(x2+4x+5)  dx by completing the square and using the substitution x+2=tan(z). Hint: sin2z + cos2z = 1.

2. Suppose ò0 1f(t)  dt = 3. Calculate each of the following integrals:

(a) ò0 5 f(2t)  dt

(b) ò0 1 f(1-t)  dt

(c) ò 01.5 f(3-2t)  dt

Project 7.3

Evaluate the following integrals by using integration by parts.

1. (a) ò r (lnr)2   dr

(b) ò 01 u arcsin u2   du

2. Let f be a twice differentiable function such that f(0)=6, f(1)=5 and f'(1)=2. Evaluate the integral ò x f''(x)   dx

Project 7.4

Evaluate the following integrals. You can use an integral table if necessary. 1. ò e-ctsin kt   dt

2.ò z3/(z-5)   dz

3. ò sin2(2x) cos3(2x)   dx

4. ò 101-x   dx

Assignments || Math Dept