Midterm 1: Sample Test
Math 1220-004
Fall 2014
Warnings:
(1) Please bring your student ID with you during midterm 1.
(2) No references or calculators can be used for midterm 1.
Things you need to remember:
(1) Definitions, rules and identities of exponentials, logarithms, trigs and
inverse trigs, hyperbolic functions (inverse hyperbolic are not required).
(2) The list on the first page of chapter 7 notes.
(3) Trig identities. (can be found on web page)
Sample Test
(1) Solve for x:
Z
x
1
3
1
dt = 2
t
Z
1
x
1
dt
t
(2) Let f (x) = x5 + 2x3 + 4x + 3
(a) Prove that f has an inverse, which is denoted by f −1 .
(b) Evaluate (f −1 )0 (10).
(3) Find the inverse functions of:
(a)
f (x) =
x3 + 4
x3 + 1
g(x) = (
x+2 5
)
x+1
(b)
(4) Differentiate the following expressions with respect to x:
(a)
x
y = xx = x(x
x)
(b)
3 +1)
(2 sin x + x2 )tan (x
(c)
(ln x2 )2x+3
(5) Given
1
lim (1 + x) x = e
x→0
compute the following:
(a)
1
lim (1 + 2x) x
x→0
(b)
1
lim (1 − x) x
x→0
(c)
3
lim (1 + x) x
x→0
(6) A bacterial population grows at a rate that is proportional to its size.
Initially, it is 10,000 and after ten days it is 20,000. The equation you
should consider is : y = y0 ekt
(a) Verbally describe the meanings of y0 and k.
(b) What is the population after 25 days?
(c) How long to double the population?
(7) Prove the following identities.
(a)
sin(arccos x) =
p
1 − x2
cos(arcsin x) =
p
1 − x2
(b)
(c)
sec(arctan x) =
p
1 + x2
(8) Prove the following identities.
(a)
(arcsin x)0 = √
1
1 − x2
(b)
(arccos x)0 = − √
(c)
(arctan x)0 =
1
1 − x2
1
1 + x2
(9) Using integration by parts, find:
(a)
Z
x2 cos xdx
(b)
(c)
Z
ln x
dx
x2014
Z
y arctan ydy
(d)
Z
x cosh xdx
(e)
Z
ln2 x801 dx
(f)
Z
t(t − 1)926 dt
(g)
Z
1
arctan dt
t
(10) Find the antiderivative of the following:
(a)
6x + 3
+x−5
x2
(b)
ex+2
+1
ex+3
(c)
(d)
−1
x + x(ln x)2
2014x
25 + 20142x