assignments

Assignment 1

• Explain the pun:

• Graph the following functions:
  • \( f(x) = \tan x \)
  • \( g(x) = |x| + x \)
  • \( h(x) = | \sin x | \)
• Find the exact value of \( \sec \tan^{-1} 2 \) (without using a calculator).
• Sketch a rough graph of the outdoor temperature as a function of time during one day. (Specify the time of year!)
• If \( f(x) = 1/x \), \( g(x) = x^3 \) and \( h(x) = x^2 + 2 \) find:
  • \( f+g+h \),
  • \( fgh \),
  • \( f \circ g \circ h \).
• Suppose \( g \) is an even function and let \( h = f \circ g \). Is \(h\) always an even function? Same question with "even" replaced by "odd" throughout.

Assignment 2

• Sketch the graph of an example of a function \( f \) that satisfies all of the following conditions:
  • \( \lim_{x \rightarrow 0^-} f(x) = 1 \)
  • \( \lim_{x \rightarrow 0^+} f(x) = -1 \)
  • \( \lim_{x \rightarrow 2^-} f(x) = 0 \)
  • \( \lim_{x \rightarrow 2^+} f(x) = 1 \)
  • \( f(2) = 1 \)
  • \( f(0) \) is undefined.
• Evaluate \( \lim_{t \rightarrow 2} \frac{t^2 + t + 6}{t^2 - 4} \).
• Is there a number \( a \) such that \( \lim_{x \rightarrow -2} \frac{3x^2+ ax + a + 3}{x^2+x-2} \) exists? If so, find the value of \( a \) and the value of the limit.
• Use the \( \epsilon,\delta \) definition of a limit to show that \( \lim_{x \rightarrow 0} x^3 = 0 \).
• Use the Intermediate Value Theorem to show that \(x^3 - 3x + 1 = 0\) has a solution in the interval \(0,1\).

Assignment 3

• Prove that the derivative of an odd function is an even function. Prove that the derivative of an even function is an odd function.
• Use the definition of a derivative to find the derivative of \(f(x) = \frac{x+1}{x-1}\).
• If \(f\) is a differentiable function and \(g(x) = xf(x) \), use the definition of a derivative to show that \( g'(x) = xf'(x) + f(x) \).
• Find the points on the curve \( y = x^3 - x^2 - x + 1 \) where the tangent is horizontal.
• Find equations of the tangent lines to the curve \( y = \frac{x-1}{x+1} \) that are parallel to the line \( x-2y=2\).
• Choose a question from the applications section on a topic that interests you. Answer it.

Assignment 4

• Differentiation drill: get /really/ good at applying the differentiation rules. How you want to demonstrate that you've done so is up to you. My recommendation is that you do lots and lots of odd-numbered problems from the book and check that you've done them correctly. Edit: And logarithm/exponential problems too; Chapter 6 of this book is a good source.

Assignment 5

• Consider \( 2(x^2 + y^2)^2 = 25(x^2-y^2) \). Find the equation of the tangent line at the point \( (3,1) \). Find all the places where the tangent is horizontal. Find all the places where the tangent is vertical. Sketch the graph.
• If \( f(x) = \sqrt{2x+3} \), find \(f''(x) \).
• Water is leaking out of an inverted conical tank at a rate of 10,000cm^3/min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6m and the diameter at the top is 4m. If the water level is rising at a rate of 20cm/min when teh height of the waters is 2m, find the rate at which water is being pumped into the tank.
• The minute hand on a watch is 8mm long and the hour hand is 4mm long. How fast is the distance between the tips of the hands changing at one o'clock?
• Choose a question from the "Problems Plus" section at the end of the chapter. Answer it.