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MA 140 - Review for Final Exam - Spring 2009
- Using the deÖnition of the deÖnite integral, evaluate
R 5
3
2 x^2 + x 4
dx.
- 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.
- Evaluate: lim n!
X^ n
i=
2 i n
n
: Also, write a deÖnite integral that this expression is equal to.
- 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.
- Evaluate:
R (^) x (^3) 2 x (^2) +x 1 p (^3) x dx
- Evaluate:
R
sec x(tan x sec x) dx
- Evaluate:
R 3
3 p^ x^2 sin^ x cos^2 x+1 dx
- Evaluate:
R 1
0
dx 1+x^2
- Evaluate:
R (^) x (^3) 4 x (^2) + p (^3) x dx
- If g (x) =
R (^) sec x e^2 x
p x^4 + 5 dx, Önd g^0 (x).
- If F (x) =
R (^3) x^2 15 sin
4 t (^2) 7 t^ dt; Önd F 0 (x) :
- Let F (x) =
Z (^) tan x
3
p 5 t^3 + 7 dx. Find F 0 (x).
- 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
- 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].
- Find f (x) if f 00 (x) = 3ex^ + 5 sin x
p x; f (0) = 1; f 0 (0) = 2.
- Find f (x) if f 00 (x) = cos x, f 0 (0) = 1, and f (0) = 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].
- Using the Second Derivative Test, Önd all relative extrema of the function f (x) = x + cos x on the interval (0; 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.)
- 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)
- 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?
- 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].
- 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
- 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.
- Find the absolute maximum and absolute minimum of the function f (x) =
x^2 x 2
on the interval [3; 7].
- 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)
- 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] :
- 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.
- Find the absolute maximum and absolute minimum of f (x) = x^3 + x^2 21 x 2 on the interval [ 4 ; 2] :
- Find the critical numbers of the function F (x) = x^4 =^5 (x 4)^2.
- Use a linearization or di§erentials to approximate
p 24 : 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?
- The graph below is the derivative f 0 (x) of a function f (x).
