Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Review for Final Exam - Calculus I - Spring 2009 | MA 140, Exams of Calculus

Material Type: Exam; Class: Calculus I; Subject: Mathematics; University: Millikin University; Term: Spring 2009;

Typology: Exams

Pre 2010

Uploaded on 08/04/2009

koofers-user-dcv
koofers-user-dcv 🇺🇸

10 documents

1 / 3

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
MA 140 - Review for Final Exam - Spring 2009
1. Using the de…nition of the de…nite integral, evaluate R5
32x2+x4dx.
2. Using nequal subintervals and the right-hand endpoint of each subinterval, use the de…nition of the de…nite
integral to compute R2
1(x2+ 1) dx.
3. Evaluate: lim
n!1
n
X
i=1 1 + 2i
n32
n:Also, write a de…nite integral that this expression is equal to.
4. A radar gun was used to record the speed of a runner at the times given in the table.
t v (t)t v (t)
0 0 3.0 10.51
0.5 4.67 3.5 10.67
1.0 7.34 4.0 10.76
1.5 8.86 4.5 10.81
2.0 9.73 5.0 10.81
2.5 10.22
(a) Use 10 subintervals and right endpoints to estimate the distance the runner covered during those 5 seconds.
(b) Use 10 subintervals and left endpoints to estimate the distance the runner covered during those 5 seconds.
(c) Use 5 subintervals and midpoints to estimate the distance the runner covered during those 5 seconds.
5. Evaluate: Rx32x2+x1
3
pxdx
6. Evaluate: Rsec x(tan xsec x)dx
7. Evaluate: R3
3
x2sin x
pcos2x+1 dx
8. Evaluate: R1
0
dx
1+x2
9. Evaluate: Rx34x2+5
3
pxdx
10. If g(x) = Rsec x
e2xpx4+ 5 dx, …nd g0(x).
11. If F(x) = R3x2
15 sin4t27tdt; nd F0(x):
12. Let F(x) = Ztan x
3
5
pt3+ 7 dx. Find F0(x).
13. The graph of y=f(x)is given below. Use it to compute: (a) R3
4f(x)dx (b) R4
1f(x)dx (c) R4
4f(x)dx (d)
R4
4jf(x)jdx
14. Let f(x) = x2x2. Find: (a) R3
2f(x)dx; and (b) The area of the region that is between the graph of f(x)and the
x-axis on the interval [2;3].
pf3

Partial preview of the text

Download Review for Final Exam - Calculus I - Spring 2009 | MA 140 and more Exams Calculus in PDF only on Docsity!

MA 140 - Review for Final Exam - Spring 2009

  1. Using the deÖnition of the deÖnite integral, evaluate

R 5

3

2 x^2 + x 4

dx.

  1. Using n equal subintervals and the right-hand endpoint of each subinterval, use the deÖnition of the deÖnite integral to compute

R 2

1 (x

(^2) + 1) dx.

  1. Evaluate: lim n!

X^ n

i=

2 i n

n

: Also, write a deÖnite integral that this expression is equal to.

  1. A radar gun was used to record the speed of a runner at the times given in the table. t v (t) t v (t) 0 0 3.0 10. 0.5 4.67 3.5 10. 1.0 7.34 4.0 10. 1.5 8.86 4.5 10. 2.0 9.73 5.0 10. 2.5 10.

(a) Use 10 subintervals and right endpoints to estimate the distance the runner covered during those 5 seconds. (b) Use 10 subintervals and left endpoints to estimate the distance the runner covered during those 5 seconds. (c) Use 5 subintervals and midpoints to estimate the distance the runner covered during those 5 seconds.

  1. Evaluate:

R (^) x (^3) 2 x (^2) +x 1 p (^3) x dx

  1. Evaluate:

R

sec x(tan x sec x) dx

  1. Evaluate:

R 3

3 p^ x^2 sin^ x cos^2 x+1 dx

  1. Evaluate:

R 1

0

dx 1+x^2

  1. Evaluate:

R (^) x (^3) 4 x (^2) + p (^3) x dx

  1. If g (x) =

R (^) sec x e^2 x

p x^4 + 5 dx, Önd g^0 (x).

  1. If F (x) =

R (^3) x^2 15 sin

4 t (^2) 7 t^ dt; Önd F 0 (x) :

  1. Let F (x) =

Z (^) tan x

3

p 5 t^3 + 7 dx. Find F 0 (x).

  1. The graph of y = f (x) is given below. Use it to compute: (a)

R 3

4 f^ (x)^ dx^ (b)^

R 4

1 f^ (x)^ dx^ (c)^

R 4

R ^4 f^ (x)^ dx^ (d) 4 4 jf^ (x)j^ dx

  1. Let f (x) = x^2 x 2. Find: (a)

R 3

2 f^ (x)^ dx;^ and (b) The area of the region that is between the graph of^ f^ (x)^ and the x-axis on the interval [ 2 ; 3].

  1. Find f (x) if f 00 (x) = 3ex^ + 5 sin x

p x; f (0) = 1; f 0 (0) = 2.

  1. Find f (x) if f 00 (x) = cos x, f 0 (0) = 1, and f (0) = 2.
  2. Let f (x) = x^4 8 x^3 2 x^2 + 120x 6. Then, f 0 (x) = 4x^3 24 x^2 4 x + 120 = 4(x + 3)(x 2)(x 4). Find the absolute maximum and absolute minimum of f (x) on the interval [-5, 3].
  3. Using the Second Derivative Test, Önd all relative extrema of the function f (x) = x + cos x on the interval (0; 4 ).
  4. A rectangular package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 108 inches. Find the dimensions of the package of maximum volume that can be sent. (Assume the cross section is square.)
  5. Let f (x) =

x^3 x^2 + 1

. Then, f 0 (x) =

x^2 (x^2 + 3) (x^2 + 1)^2

, and f 00 (x) =

2 x(3 x^2 ) (x^2 + 1)^3

.. Be sure to include the domain, the x-intercept(s) and y-intercept, any horizontal and vertical asymptotes, intervals of increase and decrease, any relative extrema, intervals of positive and negative concavity, and points of ináection. Using this information, sketch a graph of f (x)

  1. An open box with a square base is to be constructed from a piece of cardboard 12 inches on a side by cutting out a square from each corner and turning up the sides. Express the volume V of the box as a function of the length of the side of the square cut from each corner. Find the length of the side of the square cut out that maximizes the volume of the box. What is the maximum volume?
  2. Let f (x) = 2x^3 + 3x^2 12 x + 4. Then, f 0 (x) = 6x^2 + 6x 12 = 6 (x + 2) (x 1). Find the absolute maximum and absolute minimum of f (x) on the interval [0; 3].
  3. Evaluate: (a) lim x! 0 (cos 3x)^5 =x^ (b) lim x! 0

x

csc x

(c) lim x! 0 +

x^2 cot x (d) lim x! 2

x^2 4

x x 2

(e) lim x! 0

sin^2 x

x^2

(f) lim t!

2 t 3 2 t + 5

t (g) lim x! 0 ^

(1 + 2x)^1 =(3x)^ (h) lim x! 1 +

x x 1

ln x

  1. Let f (x) =

x^2 + 3

. Then, f 0 (x) =

12 x (x^2 + 3)^2

and f 00 (x) =

36(x^2 1) (x^2 + 3)^3

. Find the intervals where f (x) is increasing, decreasing, concave up, and concave down. Also, Önd all relative extrema and ináection points.

  1. Find the absolute maximum and absolute minimum of the function f (x) =

x^2 x 2

on the interval [3; 7].

  1. Sketch a graph of a function that satisÖes all of the given conditions: (a) f (0) = 0 (b) f 0 (2) = f 0 (1) = f 0 (9) = 0 (c) lim x! f (x) = 0 (d) lim x! 6 f (x) = 1 (e) f 0 (x) < 0 on (1; 2), (1; 6), and (9; 1 ) (f) f 0 (x) > 0 on ( 2 ; 1) and (6; 9) (g) f 00 (x) > 0 on (1; 0) and (12; 1 ) (h) f 00 (x) < 0 on (0; 6) and (6; 12)
  2. Let f (x) = x^3 9 x^2 + 24x 3 : Find the absolute maximum and absolute minimum of f (x) on the interval [ 1 ; 3] :
  3. Let f (x) =

x^2 2 x + 5

: Then, f 0 (x) =

2 x (x + 5) (2x + 5)^2

and f 00 (x) =

(2x + 5)^3

: Find the intervals of increase and decrease, any

relative extrema, the intervals of concavity, and any ináection points.

  1. Find the absolute maximum and absolute minimum of f (x) = x^3 + x^2 21 x 2 on the interval [ 4 ; 2] :
  2. Find the critical numbers of the function F (x) = x^4 =^5 (x 4)^2.
  3. Use a linearization or di§erentials to approximate

p 24 : 1.

  1. Answer the following questions about the proof of the Mean Value Theorem:

(a) What does the equation y =

f (b) f (a) b a

(x a) + f (a) represent?

(b) We deÖne g(x) = f (x) y. Explain why g(a) = 0 = g(b). (c) What theorem do we apply to g(x)? Why is that theorem applicable here?

  1. The graph below is the derivative f 0 (x) of a function f (x).