MATH 211 Find the Second Derivative of The Function Question

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Review for Exam 1
p.93 19, 25, 27;
p.186 Review Exercises: #1,3,4,5,7,8, 15, 17, 19, 20, 27a&b, 35
186
CHAPTER 2 Differentiation: Basic Concepts
2-84
Checkup for Chapter 2
-7
CHAPTER SUMMARY
c. y
1. In each case, find the derivative
dy
dx
5
a. y = 3×4 – 4V8+
x2
b. y = (3x – x + 1)(4 – x)
5×2 – 3x + 2
1 – 2x
d. y = (3 – 4x + 3x)3/2
2. Find the second derivative of the function
f(t) = (21 + 1)?
3. Find an equation for the tangent line to the curve
y = x2 – 2x + 1 at the point where x = -1.
4. Find the rate of change of the function
with respect to x when x = 1.
1 – 5x
5. PROPERTY TAX Records indicate that x years
after the year 2010, the average property tax on a
four-bedroom home in a suburb of a major city
was T(x) = 3x² + 40x + 1,800 dollars.
a. At what rate is the property tax increasing with
respect to time in 2013?
b. At what percentage rate is the property tax
increasing in 2013?
6. MOTION ON A LINE An object moves along
a line in such a way that its position at time t is
given by s(t) = 27 – 3x² + 2 for 1 2 0.
a. Find the velocity v(t) and acceleration aft) of the
object.
b. When is the object stationary? When is it
advancing? Retreating?
c. What is the total distance traveled by the object
for 0 51 522
7. PRODUCTION COST Suppose the cost
of producing x hundred units of a particular
commodity is C(x) = 0.04×2 + 5x + 73 thousand
dollars. Use marginal cost to estimate the cost of
producing the 410th unit. What is the actual cost
of producing the 410th unit?
8. INDUSTRIAL OUTPUT At a certain factory,
the daily output is Q = 50023/4 units, where L
denotes the size of the labor force in worker-
hours. Currently, 2,401 worker-hours of labor are
used each day. Use calculus (increments) to
estimate the effect on output of increasing the size
of the labor force by 200 worker-hours from its
current level.
9. PEDIATRIC MEASUREMENT Pediatricians
use the formula S = 0.20290425 to estimate the
surface area S (m²) of a child 1 meter tall who
weighs w kilograms (kg). A particular child
weighs 30 kg and is gaining weight at the rate
of 0.13 kg per week while remaining 1 meter tall.
At what rate is this child’s surface area changing?
10. GROWTH OF A TUMOR A cancerous tumor
is modeled as a sphere of radius r сm.
a. At what rate is the volume V = for changing
with respect to r when r=0.75 cm?
b. Estimate the percentage error that can be
allowed in the measurement of the radius r to
ensure that there will be no more than an 8%
error in the calculation of volume.
f(x) = x+1
(1 – 3x)
1 – 2x
3x + 2
13. y =
24. Use the chain rule to find for the given value
of x.
dx
a. y = u-u?; u = x – 3; for x = 0
34
b. y =
,u = Vx- 1; for x
U +1
U-12
In Exercises 14 through 17, find an equation for the
tangent line to the graph of the given function at the
specified point.
14. f(x) = x2 – 3x + 2; x = 1
4
15. f(x)
; x = 1
=
x – 3
25. Use the chain rule to find for the given value
dx
of x.
a. y = x – 4u? + 5u + 2; u = x + 1; for x = 1
b. y = Vu, u = x² + 2x – 4; for x = 2
26. Find the second derivative:
a. f(x) = 6x – 4x + 5×2 – 2x + =
1
=
x=0
x +1
X
=
16. f(x)
17. f(x) = x2 + 5; x = -2
18. In each of these cases, find the rate of change of
f(t) with respect to t at the given value of t.
a. f(t) = p – 41+51V1 – 5 at t = 4
212 – 5
b. f(1) =
at t = -1
1 – 31
19. In each of these cases, find the rate of change of
f(t) with respect to t at the given value of t.
a. f(t) = + (1 – 1) at t=0
b. f(t) = (1 – 3+ + 6)1/2 at t = 1
20. In each of these cases, find the percentage rate of
change of the function f(t) with respect to t at the
given value of t.
a. f(t) = 72 – 3t+ Vt at t = 4
b. f(t)
21. In each of these cases, find the percentage rate of
change of the function f(t) with respect to t at the
given value of t.
a. f(0) = f(3 – 21) at 1 = 1
1
b. f(t) =
2
b. z =
1 +r?
c. y = (3x + 2)
27. Find the second derivative:
a. f(x) = 4x – 3x
b. f(x) = 2x(x + 4)
X-1
c. f(x)
(x + 1)2
dy
28. Find by implicit differentiation.
dx
a. 5x + 3y = 12
b. (2x + 3y) = x +1
dy
29. Find
dx by implicit differentiation.
a. xy = 1
b. (1 – 2xy) = x + 4y
=
1
=
att = 4
t – 3
att = 0
t +1
35. POPULATION GROWTH Suppose that a
5-year projection of population trends suggests
that 1 years from now, the population of a certain
community will be P thousand, where
P(t) = -27° + 91° +8t + 200
a. At what rate will the population be growing
3 years from now?
b. At what rate will the rate of population growth be
changing with respect to time 3 years from now?
In Exercises 19 through 32, either find the given limit
or show it does not exist. If the limit is infinite, indicate
whether it is + or -.
x2 + x-2
x2 – 3x
19. lim
20. lim
x-1
1-2 X+1
X-8
1
21. lim
22. lim
1-2 2 – x
x
23. lim 2 –
X
3)
24. lim 2 +
60
(2+3)
(1-3)
25.
lim
26. lim
20
x – 3x + 5
28. lim
2x + 3
+
—r? + 5
*4 + 3x² – 2x + 7
3r2
lim
+x+1
1 1
1+
1+-+
x2
lim
+-x2 + 3x – 1
27.
29.
30.
x(x – 3)
lim
— 7 – r?
1
1
31.
lim x
32.
lim
10
-V:(+3)
x

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