Q2. Find Q3. Find (x): f(x) = x ln x Q4. Find : y = sin e5x Q5. Find ( y): y = Q6. Differentiate: y = Q7. Find ′ : = cos–1 x
Q8. If (5) = –3, what can you conclude about v(t) at t = 5?
Q9. For Figure 4-7f, sketch the graph of .
Figure 4-7f
Q10. If (7) = 4, which of these can you conclude about u(x) at x = 8?
A. u is continuous at x = 8. B. u is differentiable at x = 8. C. u has a limit as x approaches 8. D. All of A, B, and C
E. None of A, B, and C
164 © 2005 Key Curriculum Press Chapter 4: Products, Quotients, and Parametric Functions
Quick Review
PROPERTY: Second Derivative of a Parametric Function
If x = u and y = v, where u and v are twice-differentiable functions of t, then
where the derivatives of u and v are with respect to t.
reciprocal of dx/dt, you can divide by dx/dt, which equals 3t2, making the
For Problems 1 and 2, find
the equations x = 2 + t y = 3 – t2
a. Make a table of values of x and y for each integer value of t from –3 through 3. b. Plot the graph of this function on graph
paper, using the points found in part a. c. Find dy/dx when t = 1. Show that the line
through the point (x, y) from part a, with slope dy/dx, is tangent to the graph at that point.
d. Eliminate the parameter t and show that the resulting Cartesian equation is that of a parabola.
e. Find dy/dx by direct differentiation of the equation in part d. Show that the value of dy/dx calculated this way is equal to the value you found in part c using the parametric chain rule.
4. Semicubical Parabola Problem: A parametric function has the equations
x = t2
y = t3
a. Make a table of values of x and y for each integer value of t from –3 through 3. b. Plot the graph of this function on graph
paper, using the points found in part a. c. Find dy/dx when t = 1. Show that the line
through the point (x, y) from part a, with slope dy/dx, is tangent to the graph at that point.
d. Eliminate the parameter t. Find y in terms of x. From the result, state why this graph is called a semicubical parabola.
e. Find dy/dx by direct differentiation of the equation in part d. Show that the value of dy/dx calculated in this way is equal to the value you found in part c by using the parametric chain rule.
5. Ellipse Problem: The ellipse in Figure 4-7g has the parametric equations
x = 3 cos t y = 5 sin t
Figure 4-7g
a. Confirm by grapher that these equations give the graph in Figure 4-7g.
b. Find an equation for dy/dx.
c. Evaluate the point (x, y) when t = /4, and find dy/dx when t = /4. On a copy of Figure 4-7g, draw a line at this point (x, y) that has slope dy/dx. Is the line tangent to the graph?
d. Determine whether this statement is true or false: When t = /4, the point (x, y) is on a line through the origin that makes a 45-degree angle with the x- and y-axes. e. Use your equation for dy/dx from part b to
find all the points where the tangent line is vertical or horizontal. Show these points on your graph.
f. Eliminate the parameter t and thus confirm that your graph actually is an ellipse. This elimination can be done by cleverly applying the Pythagorean property for sine and cosine.
Section 4-7: Derivatives of a Parametric Function © 2005 Key Curriculum Press 165 and .
1. x = t4
y = sin 3t 2. x = 6 ln t
y = t3
6. Astroid Problem: The star-shaped curve in
Figure 4-7h
a. Confirm by grapher that these equations give the graph in Figure 4-7h.
b. Find an equation for dy/dx.
c. Evaluate the point (x, y) when t = 1, and find dy/dx when t = 1. On a copy of Figure 4-7h, draw a line at this point (x, y) that has slope dy/dx. Is the line tangent to the graph? d. At each cusp, dy/dx has the indeterminate
form 0/0. Explain the difference in behavior at the cusp at the point (8, 0) and at the cusp at the point (0, 8).
e. Eliminate the parameter t. To do this transformation, solve the two equations for the squares of cos t and sin t in terms of x and y, then use the Pythagorean property for sine and cosine.
7. Circle Problem: A parametric function has the equations
x = 6 + 5 cos t y = 3 + 5 sin t
a. Plot the graph of this function. Sketch the result.
b. Find an equation for dy/dx in terms of t. c. Find a value of t that makes dy/dx equal zero. Find a value of t that makes dy/dx infinite. Show a point on the graph for which dy/dx is infinite. What is true about dx/dt
and about dy/dt at a point where dy/dx is infinite?
d. Eliminate the parameter t. To do this, express the squares of cosine and sine in terms of x and y, then apply the
Pythagorean property for sine and cosine. e. From the equation in part d, you should be
able to tell that the graph is a circle. How can you determine the center and the radius of the circle just by looking at the original equations?
8. Line Segment Problem: Plot the graph of the parametric function
x = cos2 t
y = sin2 t
Show that dy/dx is constant. How does this fact correspond to what you observe about the graph? Confirm your observation by eliminating the parameter to get an
xy-equation. Describe the difference in domain and range between the parametric function and the xy-equation.
9. Deltoid Problem: The graph shown in Figure 4-7i is called a deltoid. The parametric function of this deltoid is
x = 2 cos t + cos 2t y = 2 sin t – sin 2t
Figure 4-7i
a. Confirm by grapher that these equations give the deltoid in Figure 4-7i.
b. Find an equation for dy/dx in terms of t. c. Show that at two of the cusps, the tangent
line is neither horizontal nor vertical, yet the 166 © 2005 Key Curriculum Press Chapter 4: Products, Quotients, and Parametric Functions
Figure 4-7h is called an astroid. Its parametric equations are
x = 8 cos3 t
derivative dy/dx fails to exist. Find the limit of dy/dx as t approaches the value at the cusp in Quadrant II.
Agnesi (1718–1799), has the equations x = 2a tan t
y = 2a cos2 t
where a is a constant.
a. Figure 4-7j shows a curve for which a = 3.
Confirm by grapher that these equations, for a = 3, give the graph in Figure 4-7j.
Figure 4-7j
b. Find dy/dx in terms of t.
c. Eliminate the parameter to get an equation for y in terms of x.
d. Differentiate the equation in part c to get an equation for dy/dx in terms of x.
e. Show that both equations for dy/dx give the same answer at t = /4, and that a line
through the point where t = /4 with this
value of dy/dx as its slope is tangent to the curve.
11. Involute Problem: A string is wrapped around a circle with radius 1 in. As the string is
unwound, its end traces a path called the involute of a circle (Figure 4-7k). The parametric equations of this involute are
x = cos t + t sin t y = sin t – t cos t
where t is the number of radians from the positive x-axis to the radius drawn to the point of tangency of the string.
Figure 4-7k
a. Use your grapher to confirm that these parametric equations give the graph shown in Figure 4-7k.
b. Find dy/dx in terms of t. Simplify as much as possible.
c. Show that the value you get for dy/dx at t = is consistent with the graph.
12. Clock Problem: A clock sits on a shelf close to a wall (Figure 4-7l). As the second hand turns, its distance from the wall, x, and from the shelf, y, both in centimeters, depend on the number of seconds, t, since the second hand was pointing straight up.
Figure 4-7l
a. Write parametric equations for x and y in terms of t.
b. At what rates are x and y changing when t = 5 s?
c. What is the slope of the circular path traced by the second hand when t = 5 s?
d. Confirm that the path really is a circle by finding an xy-equation.
Section 4-7: Derivatives of a Parametric Function © 2005 Key Curriculum Press 167 10. Witch of Agnesi Problem: The Witch of Agnesi,
13. Pendulum Experiment: Suspend a small mass from the ceiling on the end of a nylon cord. Place metersticks on the floor, crossing them at the point below which the mass hangs at rest, as shown in Figure 4-7m. Determine the period of the pendulum by measuring the time for 10 swings. Then start the pendulum in an elliptical path by pulling it 30 cm in the x-direction and pushing it sideways just hard enough for it to cross the y-axis at 20 cm. Write parametric equations for the path this
pendulum traces on the floor. Predict where the pendulum will be at time t = 5 s and place a coin on the floor at that point. (Lay the coin on top of a ruler tilted at an angle corresponding to the slope of the path at that time.) Then set the pendulum in motion again. How close do your predicted point and slope come to those you observe by experiment?
Figure 4-7m
The Foucault pendulum at the Griffith Observatory in Los Angeles, California
14. Spring Problem: Figure 4-7n shows a “spring” drawn by computer graphics. Find equations for a parametric function that generates this graph. How did you verify that your equations are correct? Use your equations to find values of x and y at which the graph has interesting features, such as horizontal or vertical tangents and places where the graph seems to cross itself.
Figure 4-7n
15. Lissajous Curves: You can make a pendulum swing with different periods in the x- and y-directions. The parametric equations of the path followed by the pendulum can have the form
x = cos nt y = sin t
where n is a constant. The resulting paths are called Lissajous curves, or sometimes Bowditch curves. In this problem you will investigate some of these curves.
a. Figure 4-7o shows the Lissajous curve with the parametric equations
x = cos 3t y = sin t
Use your grapher to confirm that these equations generate this graph.
Figure 4-7o
b. Plot the Lissajous curve with the equations x = cos 4t
y = sin t
Sketch the resulting curve. In what way do the curves differ for n = 3 (an odd number)
and for n = 4 (an even number)?
c. Sketch what you think these curves would look like. Then plot the graphs on your grapher. Do they confirm or refute your sketches?
i. x = cos 5t
y = sin t
ii. x = cos 6t
y = sin t
d. What two familiar curves are special cases of Lissajous curves when n = 1 and n = 2?
Bowditch on the Internet or other source. When and where did they live? Give the sources you used.