Chapter 2: Worksheet 1 Jack K. Cohen Colorado School of Mines
Introduction to Slopes and Tangents
Suggested Problems
Section 2.1: 1, 4, 8, 14, 18, 20, 28, 32, 34.
Figure:
Ups and downs in a balloon.
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- Figure 1 shows the time history y = H(t) of a balloon flight for
0≤t≤75. Note that the curve is not necessarily related to the spatial path of the balloon, since the horizontal axis is time t, not horizontal distance x.
- There are three local minimum points on the graph (one is at the start of the flight), label them a, b and c and estimate these values as well as
H(a), H(b), H(c). Do the same for the three local maximum points, A, B, C.
- Give the best definition you can for the concept local minimum point introduced so casually in the previous question.
- Since our curve is smooth, we can intuitively assign a slope at each point of it—except for maybe the endpoints. Using the letters
a, b, c, A, B, C introduced above, give the intervals where the slope is positive. Are these the same as the intervals where the function is increasing?
- Consider the interior extrema (i.e. maximum points and minimum points). What is the value of the slope function at these points?
- We want a condition that guarantees that a point p in the interior of the domain of a smooth function gives a maximum value H(p). State such a condition in terms of the behavior of the slope near p (i.e. just to the left and right of p, not at p itself).
- When is H(t) the biggest? (The biggest value of H(t) is called the global maximum or sometimes the absolute maximum). Similarly, when does H(t) achieve its the global minimum?
- Since the independent variable is time t, the slope function here is actually velocity (right?). When is the velocity 0? When is the velocity (locally and globally) the greatest? The least? Explain your reasoning and give approximate numerical values from Figure 1.
- (2.1.36) Prove that the line tangent to the parabola y = x2 at the point
(x0, y0) intersects the x-axis at the point
(x0/2, 0). Hints: Draw a picture. No, not a few scraggly lines and curves—draw a good picture! Label your picture. Temporarily use a distinct label, say (x1, 0), for the target point on the x-axis. Now show that
x1 = x0/2 by constructing a suitable tangent line.
- (2.1.42) Write equations for the two straight lines through the point (2, 5) that are tangent to the parabola
y = 4x - x2. Does it make sense that there are two tangents through the point (2, 5)? Advice: Don't panic. Draw a picture (use Plot if you need it). Label a general point on the parabola as
(a, y(a)) = (a, 4a - a2). You can figure out the slope at the general point a and hence write an equation for the tangent line in point slope form (with a's in it all over the place). There is one fact you haven't used—use it to get an equation for a. Then solve for a and
plug back into the tangent equation.
For the really brave: replace the point (2, 5) by a general point
(x0, y0).
- (2.1.43) Find the point on the parabola y = x2 that is closest to the point (3, 0) by using the insight that for this point, the line from (3, 0) to the parabola will be normal to the parabola. Note: you have to solve a cubic equation to find the closest point—here, you can probably guess the relevant root, but if not use .
- Following up on previous problems:
- Are there always exactly two tangent lines from a general point
(x0, y0) to the parabola y = x2?
- Is there always exactly one normal line from a general point
(x0, y0) to the parabola y = x2?
- Honor Problem: In finding the ``physically sensible'' solution to the asteroid problem you may have noticed that there are a lot (infinitely many actually) of roots in the interval
[0, s = 1000]. Find the largest two roots less than s = 1000 and explain their significance. Further investigations and discussions along this line are welcome.