Download Activities - WVU Math Department

Mat
h
154
Lab
1. Look at the applet above or on Lab page 1. You can enter a function, graphed on the left, and
its derivative is calculated and graphed on the right. Sliders at the bottom right let you adjust
lower and upper limits of integration. There is a drop down menu at the top to consider different
examples. In the first screen, the area of the rectangular region is reported as 160. What are the
dimensions of the rectangle?
a. 16*10
b. 20*8
c. 32*5
d. 40*4
e. 80*2
Answer: _____
2. You can adjust the sliders or enter numbers directly in the a and b boxes. What happens
when a=b, corresponding to a definite integral where the upper and lower limits are the same?
a. The area is zero because f(a)=f(b).
b. The area is zero because f(a)=f(b)=0.
c. The area depends on the particular value used for a and b
d. The area is 20
e. The area is 160
Answer: _____
3. Interpret the two graphs as illustrating the Fundamental Theorem of Calculus. What is the
x
derivative of the function of x defined by
a. 20
b. 20 x
c. 20 t
d. 20 b - 20 a
Answer: _____
 20 dt
0
4. What happens to the area when b<a ?
a. It is the same as if b>a
b. It is half the size as when b>a
c. It is double the size as when b>a
d. It is the negative of the value reported for b>a
e. It is the reciprocal of the value reported for b>a
Answer: _____
5. Go to the second function in the drop down menu, a parabola. Find the area when a=0 and
b=4. Find the area when a=-4 and b=4. (The E-14 indicates exponential notation, so the value in
the second window is essentially 0). What property of the parabola accounts for the reported area
when a=-4 and b=4?
a. The parabola has vertex at the origin
b. The parabola has exactly one x intercept
c. The parabola has symmetry with respect to the x axis
d. The parabola has symmetry with respect to the y axis
e. The parabola has symmetry with respect to the origin
Answer: _____
6. The parabola represents a quadratic function f(x) that is an antiderivative of the function f'(x)
graphed in the right window. The linear function on the right has an area bounded by the function,
vertical lines, and the x axis that can be calculated by geometry. What is the calculation for the
area on the right when a=1 and b=3, via the formula for the area of a trapezoid?
a. 2+6
b. 8*1
c. 1*8
d. (2+6)/2 * 2
e. 1+2+2+2+1
Answer: _____
7. With a=0 and b>0, the shaded region in the right hand window is a triangle. What are the
dimensions of the triangle as a function of b?
a. base b, height b
b. base b height 2*b
c. base 2*b, height b
d. base 2*b, height 2*b
e. base b, height 3
Answer: _____
8. For those dimensions of the triangle, what is the area of the triangle as a function of b?
a. 2*b
b. b^2/2
c. b^2
d. 2*b^2
Answer: _____
9. Go to the third function in the drop down menu, a cubic. Compare the area using a=0 and
b=5 with the area using a=-5 and b=0. What do you notice?
a. They are the same, by a numerical accident
b. They are the same, by symmetry of the parabola
c. They have the same absolute value but opposite signs, by the odd symmetry of the
cubic
d. They are unrelated
Answer: _____
10. Look at the fourth function in the drop down menu, a trig function. Set a=-Pi/2. What value
for b makes the area under the curve as large as possible?
a. 0
b. Pi/2
c. Pi
d. 3*Pi/2
e. 2*Pi
Answer: _____