Mat
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Lab
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1. This laboratory explores the relationship between the derivative of a function and the
derivative of the inverse function. First recall the geometric interpretation of the inverse function
graph. Go to lab page 1. It is preloaded with the equation y=sqrt(x), the square root function.
Press the button to graph it, the the "Graph inverse relation" button to graph the inverse. What is
the symmetry between the two graphs?
a. Symmetry with respect to the x axis
b. Symmetry with respect to the y axis
c. Symmetry with respect to the origin
*d. Symmetry with respect to the line y=x
e. Symmetry with respect to the line y=-x
2. What point on the red curve is symmetric to the point on the blue curve corresponding to
x=4? Give your answer as an ordered pair of numbers (a,b) without any spaces.
Correct Answer(s):
a. (2,4)
3. Now enter the function y=x^2 and graph it and its inverse. Compare this situation to the
previous one. Mark the true statements below.
*a. The blue graph is not a one-to-one function, so its inverse relation is not a function.
*b. The graphs still exhibit symmetry about the line y=x.
*c. We could recover the graph in the previous example by restricting the domain of the
blue function, hence the range of its inverse.
*d. There are exactly two points of intersection of the graphs of the function and its
inverse relation.
e. The function's graph lies in quadrants I and II, so the inverse graph lies in quadrants
III and IV.
4. Now go to lab page 2. Here is the graphing utility we have been using, but it has an extra
feature: Besides graphing two functions, it also tracks the tangent lines independently. The page
is preloaded with two functions. Observe what happens when you drag the sliders under the
graph. Note the effect on the graph and on the text boxes reporting values of x and f'(x), x and
g'(x). The point of tangency can be adjusted separately for f(x) and g(x). You can enter particular
values of x in the table on the right to see function values, and you can enter values of x in the
text boxes on the left to see values of the derivatives. Keep track of which function is which by
color (red and blue) or by label (f on top or to the left). Mark all of the following that are true
*a. On the blue curve, g'(x) is undefined for x < 0.
*b. On the blue curve, g(x) increases when x increases.
c. On the blue curve, g'(x) increases when x increases.
*d. On the red curve, when x > 0, g(x) increases as x increases.
*e. On the red curve, when x>0, g'(x) increases as x increases.
5. Consider the point (9,3) on the blue curve and its symmetrical point (3,9) on the red curve.
Compare the values of the derivatives at these points. Note that you can enter x directly in the
text boxes on the left if you don't want to use the sliders.
a. They are the same
b. One is the negative of the other
*c. They are reciprocals
d. They are negative reciprocals
6. Now go to lab page 3. f(x) is a simple cubic function. My grapher has trouble with rational
exponents, so I had to enter a more complicated version of the inverse function. Find a simpler
algebraic form of the inverse function by solving the equation x=y^3-5 for y.
a. y=x^3-5
b. y=x^3+5
c. y=x^(1/3)-5
d. y=x^(1/3)+5
e. y=(x-5)^1/3
*f. y=(x+5)^1/3
7. The grapher calculates the derivative by sampling, and is inaccurate at some points.
Consider the horizontal tangent line when x=0 on the red curve. What goes wrong with the
reported value on the symmetric point on the blue curve?
8. Now go to lab page 4. A famous pair of inverse functions is the exponential function and the
natural log function. The point (a,b) is on the exponential function graph (the red graph) if the
point (b,a) is on the log graph (the blue graph). So exp(a)=b conveys the same information as
that
a. ln(a)=b
b. a=b
*c. ln(b)=a
d. exp(a)=exp(b)
e. ln(a)=ln(b)
9. Stay on lab page 4. Use the table on the right to find f(2). Observe that f(2) =f'(2)=7.389. This
is an amazing property of exp(x). Find g'(7.389) What is the relationship between f(2) and
g(7.389)?
a. They are the same
b. They have the same absolute value, one is positive and one is negative
*c. They are reciprocals
d. They are negative reciprocals
10. Go to lab page 5. The notation for the inverse sin function in the grapher is asin, short for
arcsin, a variation of sin-1 . It is found from sin(x) by restricting the domain so that sin(x) is one to
one. What is true about the graph of asin(x) ?
*a. The graph is increasing on its domain
*b. The derivative is positive where ever it is defined
*c. asin(x) is defined for -1 < x < 1
d. asin(x) is defined for -Pi/2 < x < Pi/2
e. asin(45o)=sqrt(2)