201-NYA Final Exam | May 2009
Calculus I
1. Refer to the sketch below to evaluate the following. If a value does not exist, state in which way (+1, 1,
or \does not exist").
(a) x!lim1 f (x) =
y
6
(b) xlim
f (x) =
!0
8
(c) xlim
f (x) =
!2
7
(d) xlim
f (x) =
!3
6
(e) xlim
f (x) =
!3+
(f) f (3) =
(g) x!lim
f (x) =
+1
2.
3.
4.
5.
6.
5
b
4
b
3
r
2
-x
(h) List the values of x where f (x) is discontinuous.
2
1
1
2
3
4
Evaluate the limits. Use the symbols 1 or +1 where appropriate.
px 3
6
11
x + 3x2
6
11
x + 3x2
(a) x!lim
(c) xlim
(b) xlim
+1 3 + 14x
!9 x 9
!3 3 + 14x 5x2
5x2
3
j
xj x
sin
2
x
(e) xlim
(d) xlim
!0 tan x
!0+ x2
p
3
x + x2 + 1
Find all the vertical and horizontal asymptotes for f (x) = 2x 1 .
(a) State the denition for a function f (x) to be continuous at x = a.
(b) Using this denition, determine if the following function f (x) is continuous at x = 0. (Show your
justication
8 for your2 answer.)
< (x 1) if x < 0
2
if x = 0
f (x) =
: cos(x)
if x > 0
(c) Sketch the graph of the function y = f (x) in part (b).
Sketch (if possible) a function f that is continuous but not dierentiable at x = 0.
State a limit denition of the derivative for a function f (x).
Use the above denition to nd the derivative of f (x) = 1 2 x .
1
7. For each of the following functions, calculate the derivative dy
. You do NOT have to simplify your answers.
dx
p
7
(a) y = x7 p5 2 2 + 3x ln 2 + 3x
(b) y = x8 sec x
x
3
p
(x + 1)
(c) y = tan
(d)
y = sin(ex + e2 )
ln(3x 2)
2 + 1)
(e) y = (x + 1)(x
8. Find the equation of the tangent line to the graph of
9. Given
x3 + y 3
(a) nd
dy
dx
= 6xy + 1 :
y
= cos(x x+ 1 1) at the point where x = 1.
using implicit dierentiation;
dy
(b) evaluate dx
at (1; 0).
2
10. Find the absolute maximum and minimum values of h(x) = xx2 + 11 on the interval [ 1; 1].
6
Calculus I
201-NYA Final Exam | May 2009
11. The graph of y = f (x) is given at right.
(a) At what value(s) of x does f 0 (x) change sign?
(b) At what value(s) of x does f 0 (x) have a local
(relative) maximum?
(c) At what value(s) of x does f 0 (x) have a local
(relative) minimum?
(d) Sketch a rough graph of f 0 (x) (on the same axes).
(e) At what value(s) of x does f 00 (x) change sign?
x1
x2 x3
x4
x5
-
12. A boat is pulled into a dock by a rope attached to the bow of the boat, passing through a pulley on the dock
that is 10 meters higher than the bow of the boat. The rope is pulled in at a constant rate of 5 meters per
minute.
(a) If s is the length (in meters) of rope between the pulley and the bow of the boat, express s as a function
of the angle between the water and the rope.
(b) At what rate is the angle changing when the length s = 26 meters?
13. For the function f , given below with its derivatives:
x 2
6 2x
f (x) =
f 0 (x) =
x3
f 00 (x)
x4
= 6x x5 24
sketch the graph, identifying all intercepts, local extrema, and inection points. Specify intervals where the
graph is increasing, decreasing, concave up, and concave down. Show all your work.
14. A rectangular box has an open top, a square base, and a surface area of 147 square meters; nd the dimensions
of the box if it is to have the maximum possible volume.
15. Given that f 0 (x) = x + sin(x) , and f (0) = 3 , nd f (x) .
16. Evaluate
the following integrals:
Z Z 4
4 t5 + 5 et 1 dt
(b)
(a) x 24xx2 1 dx
e5
t5
4
Z
Z4
p
(c) sec x(tan x sec x) dx
(d) ( x + 2)2 dx
17. (a) Evaluate the Riemann sum for f (x) = sin(x), 0 sample points to be the midpoints.
Z
(b) Evaluate sin d .
1
x
, with three equal subintervals, taking the
o
18. Suppose F (x) =
Z px
0
(a) What is F (0)?
et2 dt .
(b) What is F 0(x)? (Hint: you may wish to use the
19. Find the area of the region bounded by the curve y = x
Fundamental Theorem of Calculus.)
2
2x and the x-axis.
201-NYA Final Exam | May 2009
Calculus I
Answers
(a) +1 (b) 1 (c) +1 (d) 5 (e) 4 (f) 3 (g) 8 (h) 0; 2; 3
(a) 35 (b) 167 (c) 16 (d) 2 (e) +1
VA: x = 21 HA: y = 1 and y = 2
(a) (i) f (a) exists (ii) xlim
!a f (x) exists (iii) f (a) = xlim
!a f (x).
(b) Discontinuous at x = 0 because f (0) = 2 6= 1 = xlim
!0 f (x) (c) Graph:
5. Many examples, such as the \V-shaped" absolute value graph.
f (x + h) f (x)
(if the limit exists)
6. f 0 (x) = hlim
!0
h
2
2
(2 2x) )2 (2x + 2h)) = lim
2
1 (x+h)
1 x
= hlim
f 0 (x) = lim
!0 h(1 x h)(1 x)
h!0 (1 x h)(1
h!0
h
7. (a) x6 + 54 x 7=5 + 61 x 1=2 + 3x ln 3
(b) x8 sec x tan x + 8x7 sec x
2
2
x 2) 3 tan3 (x + 1)=(3x 2)
(c) 3 tan (x + 1) sec (x + 1) ln(3
2
ln (3x 2)
(d) 21 (sin(ex + e2 )) 1=2 cos(ex + e2 ) ex
+1
(e) (x + 1)x2 +1 xx2+1
+ 2x ln(x + 1)
1.
2.
3.
4.
8. y =
1
x
4
+ 34
x)
= (1 2 x)2
9. (a) 2yy2 2xx (b) 21
10. Abs max at ( 1; 0); (1; 0) Abs min at (0; 1)
11. (a) x1 ; x3; x5 (b) x2 (c) x4 (d) /\/ zigzag shape, crossing the x axis at x1; x3; x5.
(e) x2 ; x4.
12. (a) s = 10 csc (b) 25=312
13. VA at x = 0, HA at y = 0, x-intercept at x = 2,
CP (local max) at x = 3, PI at x = 4,
increasing for x < 0 and 0 < x < 3, and decreasing for x > 3,
concave up for x < 0 and for x > 4, and concave down for 0 < x < 4.
Graph at right.
14. 7m 7m 72 m
15. f (x) = 12 x2 cos x + 4
16. (a) 16 x3 2 ln jxj + 21x + C (b) t 4 241 t6 + 5 et e15 t + C
(c) sec x tan x + C (d) 229
6
17. (a) 23 (b) 2
18. (a) 0 (b) 2epxx
19. 43 [not 43 !]
2