AP
Worksheet
Section 2.4
Find the average rate of change over each interval.
Name ________________________________________hr 4
Date ____/____/____ Score:________/
1. f ( x)  x 3  1
(a) [2, 3]
(b) [-1, 1]
average rate of change = __________
average rate of change = __________
2. f ( x)  e x
(a) [-2, 0]
(b) [1, 3]
average rate of change = __________
average rate of change = __________
3. f ( x)  cot t
(a)  4 , 3 4
(b)  6 ,  2
average rate of change = __________
average rate of change = __________
4. Accelerating from a Standstill
The figure shows the distance-time graph for a 1994 Ford Mustang Cobra accelerating from a standstill.
(a) Estimate the slopes of the secants PQ1, PQ2,
PQ3, and PQ4 arranging them in order in a table.
What is the appropriate unit for these slopes?
Secant
Slope
PQ1
PQ2
PQ3
PQ4
(b) Estimate the speed at point P.
paste diagram here
= ________%
For exercises 5 - 6, at the indicated point find (a) the slope of the curve, (b) an equation of the tangent, and (c) an equation
of the normal, and (d) Then draw a graph of the curve, tangent line, and normal line in the same square viewing window.
5. y  x 2 at x = -2
6. y 
(a)
(a)
(b)
(b)
(c)
(c)
(d)
(d)
1
at x = 2
x 1
7. Find the slope of the curve, f ( x)  x at the indicated point
(a) x = 2
(b) x = -3
slope of the curve = ______________________
slope of the curve = ______________________
Determine whether the curve has a tangent at the indicated point. If it does, give its slope. If not, explain why not.
2  2 x  x 2 , x  0
8. f ( x)  
 2 x  2, x  0
, at x = 0
 1 x , x  2
, at x = 2
,
x

2
 4
9. f ( x)   4  x
For exercises 10 & 11 (a) Find the slope of the curve at x = a. and (b) Describe what happens to the tangent at x = a as a
changes.
1
x 1
10. y  x 2  2
11. y 
(a) slope of the curve = ______________________
(b)
(a) slope of the curve = ______________________
(b)
12. An object is dropped from the top of a 100-m tower.
Its height above ground after t sec is 100- 4.9t2 m. How
fast is it falling 2 sec after it is dropped?
13. What is the rate of change of the area of a circle with
respect to the radius when the radius it r = 3 in?
14. The equation for free fall at the surface of Mars, is
s = 1.86t2 m with t in seconds. Assume a rock is dropped
from the top of a 200-m cliff. Find the speed of the rock
at t = 1 sec.
15. At what point is the tangent to f ( x)  x 2  4 x  1
horizontal?
16. Finding Tangents and Normals
(a) Find the equation in slope-intercept form for each
1
tangent to the curve y 
that has a slope of -1.
x 1
(b) Find an equation for each normal to the curve
y
1
that has a slope of 1.
x 1
17. Table 2.2 gives the amount of federal spending in
billions of dollars for national defense for several years.
(a) Find the average rate of change in spending from
1990 to 1995.
Table 2.2 National Defense Spending
Year
National Defense Spending ($ billions)
1990
1995
1999
2000
2001
2002
2003
299.3
272.1
274.9
294.5
305.5
348.6
404.9
(b) Find the average rate of change in spending from
2000 to 2001.
(c) Find the average rate of change in spending from
2002 to 2003.
(d) Let x = 0 represent 1990, x = 1 represent 1991,
and so forth. Find the quadratic regression equation
for the data and superimpose its graph on a scatter
plot of the data.
(e) Compute the average rates of change in parts (a), (b),
and (c) using the regression equation.
(f) Use the regression equation to find how fast the
spending was growing in 2003.
(g) Explain why someone might be hesitant to make
predications about the rate of change of national
defense spending based on this equation.