Assignment 1

Functions and Graphs

Review of some topics from Algebra

    1. Sketch the set of points described by |2x-3|<1
    2. Describe the set of points –2<x<7 using absolute values
    3. What is the equation for the circle of radius 3 centered at (1,-2)?
    4. Give the equations of the following lines in point-slope form.

a.        the line through (2,3) with slope –1

b.       the line through (-2,5) and (1,3)

c.        the line parallel to y=2x+6 through (1,1)

d.       the line through the origin perpendicular to the line y – 2x = 5

    1. Sketch the set of points described by x^2 +(y+3)^2 = 4

Review of some topics from Trigonometry

    1. Show on the unit circle which angles have
    1. sines of .4 .
    2. cosines of -.3
    3. tangents of 3

Functions

    1. Which of the following define y as a function of x? Why?
    1. y = x^2 + x
    2. x^2 + y^2 = 1
    3. y = 3
    4. sin(y) = x
    1. Let f(x) = sin(x) and g(x) = x^2. What is f(g(x))? What is g(f(x))? Are they the same? Plot them. (Use MAPLE and choose a reasonable x range.)
    2. Which of the following are polynomials?
    1. x^3 – x + 13
    2. (x+1)^(1/2)
    3. (x-3)^2
    4. (x+1)/(x-2)
    1. Plot the functions in exercise (9).
    2. Tell me how long this assignment took. What was hardest?

 

 

Assignment 2

Inverse Functions, Logarithms & Exponentials

Inverse functions

    1. Plot the following. (Use MAPLE.) Which have inverses? Why? Find them.
    1. x^3
    2. x^2 + x + 1
    3. (x+1)/(x-2)
    4. x – 1/x

Graphs of trig functions

    1. Plot all six trig functions. What are the periods?
    2. Plot 2*sin(3*x) – cos(3*x). What is the period? What is the amplitude?

Inverse trig functions

    1. Evaluate the following:
    1. sin(arccos(.3))
    2. cos(arctan(10))
    3. tan(arcsin(-.2))
    1. What is sin(arcsin(x))? How about arcsin(sin(x))? Explain.
    2. Let A = the angle in the second quadrant with sine = .7, and find all five other trig functions of A.

Exponentials

    1. LEARN the rules of exponents!
    2. Graph 3^x, 3^(-x), (1/3)^(x). What do you notice? Prove it!

Logarithms

    1. Assume that x = b^y. What does this say about the log base b of x? Now take the log base a of both sides of x = b^y. Derive the change of base formula.
    2. Plot log base 10 of x, log base 2 of x, and the natural log of x.

Story problems!

    1. How much interest does $1000 earn at 5% per year if it is compounded
    1. monthly?
    2. daily?
    3. hourly?
    1. If T(t) is the temperature of an object, and Troom is the room temperature, then T(t) is given by an equation of the form T(t) = Troom + [T(0) – Troom]*exp(-k*t). (T(0) is the temperature at time 0, and k is a constant.) Suppose my coffee starts out at 180 degrees Fahrenheit, and after 5 minutes it is 140 degrees. Assuming room temperature is 70:
    1. Find k
    2. Find the temperature after 15 minutes.
    3. When will the temperature be 100?
    1. Tell me how long this assignment took. What was hardest?

Assignment 3

Derivatives – slopes, tangents, "zooming"

  1. Take the function f(x) = x^2, and a point on its graph, (a,a^2) (Choose your favorite a.) Draw the lines through the points at x = a and x = a +h for h = 1, .5, .1 and then sketch the tangent line.
  2. Do the same thing with some negative h values, getting close to 0.
  3. Now try the lines from x = a-h to x = a+h.
  4. Have MAPLE simplify the expression [f(a+h) – f(a)]/h. What happens as h gets small? How does this relate to what you did above?
  5. Now define f as a function in MAPLE, and take its derivative (using either the D or diff operators.)
  6. What is the equation of the tangent line?
  7. Repeat exercises 1 – 6 with a couple of other functions.
  8. Plot a polynomial in MAPLE and choose a point on the graph. By choosing the horizontal and vertical ranges in the plot function, "zoom" in symmetrically about your point. What do you observe?
  9. Use diff or D to find the derivative of your polynomial. What is the tangent line at your point? Plot the polynomial and the tangent line on the same graph and repeat exercise 8.
  10. Repeat exercises 8 – 9 for another function. Use a trig function this time.
  11. Plot the following functions and determine just from the graph where the derivative is positive and where it is negative
  1. x^3 + 2*x^2 – x – 2
  2. sin(x)
  3. x*exp(-x)
  4. ln(x)
  1. And our usual last question, how long did this take and what was the hardest?

Assignment 4

Rates of Change

  1. The position of an object is described by s(t) = (t^2+2*t)/(t^2+t+1). Plot this function. Also, find and plot the velocity and the acceleration of the object. Discuss.
  2. Starting from ground level, a ball is thrown up at 5 m/s. When does it hit the ground? What was its average velocity? Average speed?
  3. How fast does a ball hit the ground if it is thrown upward at 6 m/s from the top of a 10 m building? What if it were thrown down at 6 m/s?
  4. What is the volume of a sphere in terms of the radius? What is the rate of change dV/dr? Does this look at all familiar? Explain
  5. Repeat the previous exercise (which I believe to be number 4) with a cube.
  6. How fast should a ball be thrown upward to just barely hit the ten foot ceiling?
  7. The thickness of the layer of lime in your water pipes grows at a constant rate. If your pipes have inside diameter .75 inch, how fast is the open cross section changing, relative to the thickness of the lime layer, when the layer is .1 inch?
  8. Use the tangent line to the curve y = x^(1/2) at some appropriate point, to approximate the square root of 3. Now use a closer point, and improve your approximation.
  9. Plot the curve and the first 2 tangent lines to approximate the root of x^3 – 2*x^2 – 2*x – 2
  10. When you measure the diameter of a circle, you find it to be 6 cm, to within 1%. How accurately can you calculate the area?
  11. Suppose a soccer ball is made of leather 1/8 inch thick. If the outside diameter is 9 inches, and the density of leather is assumed to be .64 oz/cubic inch, how much does the ball weigh?
  12. How long did this take and what was the hardest?

Assignment 5

Chain Rule

  1. Let f(x) be the square root of x, and g(x) be 1/(x^2+1). What are the range and domain of f(g(x))? Of g(f(x))?
  2. What are the derivatives of f & g? From the chain rule, what should the derivatives of the functions f(g(x)) and g(f(x)) be? Check this.
  3. A boy is flying a kite. If the kite is moving 10 ft/sec horizontally south, and is 60 feet above the ground, how fast is the string unwinding when the kite is 100 feet south of the boy?
  4. A 1.6 meter tall woman is walking past a 3 meter lamppost. If she is walking at 2 m/s directly away from the lamp, how fast is the length of her shadow changing when she is 5 m away?
  5. A lighthouse is 1 mile offshore, due west. You are on the shore, 2 miles north of the point opposite the lighthouse. If you see the flash every 15 seconds, how fast is the beam of light moving when it passes you?
  6. A trough is triangular in cross section, an isosceles triangle with sides 12 inches, and top 10 inches. The trough is 40 inches long. How fast is the depth changing if you are pumping one cubic foot per minute into the trough?
  7. In a moment of foolishness, you tie your dog to the foot of the 4 meter ladder you are climbing. When you are 1.5 m from the top, the dog leaves, at ¾ km/hr, chasing a rabbit. How fast do you hit the ground?
  8. A light is attached to a 13 inch radius bicycle tire, at a point 8 inches from the center. If Harvey rides the bicycle at 15 mph, how fast does the light move up and down at its fastest?
  9. And, as usual, how long and which was hardest?

Assignment 6

Implicit Differentiation

Parameterized functions

  1. Consider the circle x^2 + y^2 =1. Find the slope of a tangent to the circle two ways: a) by solving for y, and taking the derivative and b) by implicit differentiation.
  2. Repeat the previous problem with x = y^2.
  3. Find the derivative of exp(-x), exp(a*x), exp(f(x)).
  4. Use implicit differentiation to find the derivative of arccos(x).
  5. Use implicit differentiation to find the derivative of arctan(x).
  6. Use implicit differentiation to find the derivative of ln(x).
  7. Find the derivative of x^(1/x)
  8. Find a parametric equation for the ellipse (x^2) + (y^2)/4 = 1. Find the derivative three ways: a) solve for y, b) implicitly, and c) using the parametric form.
  9. Suppose a reflector is attached 8 inches from the center of a 24-inch diameter bike wheel. If the bike wheel rolls with out slipping, find parametric equations for the position of the reflector. Plot it.
  10. Try putting the reflector at different distances from the center and plot.
  11. And, as usual, how long and which was hardest?

Assignment 7

Max – Min Problems

 

  1. Find the critical points of x^3-2x^2+x-2 on the interval [0,5]. Identify the type of each. Plot.
  2. Find the local and global minima and maxima of x^(2/3) on the interval [-1,2]. Plot.
  3. If two numbers add to 12, what is the biggest their product can be?
  4. A poster is to have 150 square inches of printed material. If we want a 2 inch margin at the top, and 1 inch margins on the other three edges, what dimensions use the smallest amount of paper?
  5. The material for the top of a cylindrical can cost three times as much as the material for the sides and bottom. What dimensions should the can have to hold a fixed amount?
  6. An off shore oil rig is 3 km west of the shore. We wish to pump the oil to a location on the shore 5 km north of the rig. If it costs 1.8 times as much per kilometer to build an underwater pipeline as one on land, how should we design our pipeline?
  7. To make pompoms in our school colors, we will have expenses of $100 to rent the Acme PomPomPlant, and then $.25 / pompom in materials. We believe that we can sell 500 pompoms if we charge $1, and that if we charge $1.50 we will only sell 300. Assume that the number sold is a linear function of cost. How much should you charge to make the biggest profit?
  8. Strength of a rectangular beam is proportional to width times (depth)^2. What are the dimensions of the strongest beam that can be cut out of a 12 inch diameter log?
  9. A line in the first quadrant is tangent to the graph of y = 1/x^2. How can we minimize the area between the line and the axes?

Assignment 8

Max – Min Problems

  1. Imagine a function with positive second derivative. Start at x=0 with slope –2, and approximate the function with a piece of straight line. Suppose that the slope increases by one every time x increases by one, and continue approximating by pieces of straight line (do this by hand, not MAPLE). What does your sketch look like?
  2. Must a function with positive second derivative go up?
  3. Consider the function x^(4/3), on the interval [-2,2]. Find the critical point. What does the second derivative test say? Plot the function and explain.
  4. If the speed of light in water is about ¾ the speed of light in air, how much does a beam of light striking water at an angle of p /8 bend?
  5. A pyramid consists of 4 equilateral triangles around a square base. If this is to be cut and folded out of a single square piece of paper (base in the center, tops of triangles in the corners), maximize the volume.
  6. What is the longest pipe that can be carried horizontally around a corner from a 5 foot wide corridor into an eight foot wide one?
  7. We are told that if 60 grapevines are planted per acre, each vine will produce 120 lbs. of grapes, but that each additional vine per acre reduces the average yield per vine by 3 lbs. How many vines should be planted to maximize the yield per acre?
  8. The labor to tend the grape vines costs $4 per vine per season. The grapes sell for 65 cents per pound. How many vines should be planted to maximize the profit?
  9. Taxes on the land are $50 per acre. Does this change the best strategy? How?
  10. How would you build the cost of the vines into your model?

Assignment 9

  1. Approximate the area under the curve y = x^2 + x between x=1 and x=3 using 10 rectangles and (a) the left endpoint (b) the right endpoint, (c) the midpoint.
  2. Repeat problem 1 with the upper half of the circle x^2 + y^2 = 1. Compare to the exact answer.
  3. Prove that

  1. The trapezoid formula approximates an area by using a straight line on each subinterval; this give tall, skinny trapezoids. Write out a general formula for the trapezoid formula to approximate the area under f(x) from x=a, to x=b. Now for n=5 expand out the sum. What do you see?
  2. Standard last question J

 

Assignment 10

  1. Show that the fundamental theorem is true for f(x) a straight line (get the area geometrically).
  2. Find an antiderivative for (a) x^3 + 2*x^2 – x +1 (b) sqrt(x) (c) cos(x) (d) sin(x)
  3. If diff(f(x),x) = diff(g(x),x) –1, what can you say about f and g?
  4. What is the antiderivative of x^n? For what values of n is this a problem?
  5. Consider

Sketch a graph that represents this integral. Does this graph make sense at n = -1? Evaluate the integral, and choose a value for x. Plug in values of n getting closer and closer to –1. Do you approach a limit?

  1. Evaluate the integral in problem 5, for n = -1, using MAPLE. Does this agree with the limit you got at the x value you chose?
  2. Evaluate the integral of sin(x) from 0 to 2p using MAPLE. Does this make sense as an area? Sketch the graph. What do you see? Now find the actual area between sin(x) and the axis between x = 0 and x = 2p .
  3. Let H(x) = 0 for x<0 and 1 for x>0. Sketch H. What is the integral from 0 to x of H? Sketch it. How about the integral from –1 to x of H? From 2 to x?
  4. Now let G(x) be 0, for x<0; x, for 0<x<2; and 2, for x>2. Repeat the previous problem for G.
  5. Standard last question.

Assignment 11

  1. Set up the volume of a cone as an integral. Evaluate.
  2. Set up an integral to find the volume of an equilateral tetrahedron. Evaluate.
  3. Consider the solid obtained by rotating the region in the first quadrant between x=0, y=0 and y = 2 – x^2 around the x – axis. Set up the integrals for the method of shells, and the discs. Evaluate both.
  4. The wetted face of a dam is a trapezoid 40 feet wide at the top, and 30 feet wide at the bottom, at a depth of 12 feet. The face is slanted do that the bottom is five feet in from the top. Find the total hydrostatic pressure on the face.
  5. A water tank is a sphere, 3 m in diameter, 15 m above ground level. How much work does it take to fill the tank half way? All the way?
  6. Find the total work to put a one ton satellite in geo-synchronous orbit, approximately 24,000 miles up. (You may assume the earth is a sphere with radius 4000 miles. Force due to gravity is proportional to 1/distance^2, and you know what it is at distance = 4000 miles.)

Assignment 12

  1. Set up integrals to find the arc length of y = x^3 from (1,1) to (3,27) by integrating with respect to (a) x and (b) y. Evaluate them both.
  2. Set up three different integrals to find the circumference of a circle, integrating with respect to x,y and a parameter q . Which looks easiest?
  3. Find the surface area of a sphere using a surface area integral.
  4. Do the same for a cone.
  5. Find the circumference of the ellipse x^2/a^2 + y^2/b^2 = 1. Use a parametric equation.
  6. Find the surface area, and the volume, of the solid obtained by rotating the area bounded by y = (1-x^2)/3 and the x-axis about the x-axis.
  7. Standard last question.

Assignment 13

  1. Find the centroid of the region bounded by 1 – x^2 and the x – axis.
  2. Use the theorem of Pappus to find the centroid of a semicircle with out integrating.
  3. Set up and evaluate the integrals to check your answer in #2.
  4. Let g(x) = 1/f(x) and apply the product rule to f(x)*g(x). Solve for dg/dx.
  5. Use the result in the previous problem, and the product rule to get a formula for the derivative of u(x)/v(x).
  6. Use integration by parts to integrate x*exp(-2*x)
  7. Use integration by parts to integrate x^2*sin(3*x)
  8. Use integration by parts to find the area under x*ln(x) between 1 and 5.
  9. Use the substitution x = sin(t) to integrate 1/sqrt(1-x^2)
  10. Find the substitution to make 1/(x^2+x+1) = 1/(u^2+1). Integrate (using Maple) both ways.
  11. What are the three most important things to change the next time we offer this class?