Math1220 Midterm 1 Review Problems
(6.1-6.5, 6.8, 6.9, 7.1, 7.2)
1. Find the equation of the tangent line to the graph of
f
2. Find
3. Find
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
−1
 x  for

2x−1
f  x =
2x5

3
y=cos−1 (ln ( x 4 )) when x = 1.
.
dy
for each function. (Don't simplify.)
dx
y=ln cos 2  3xsin −1  3x−2
y=5x32x
y=1 x 4   1 x
y=sech cos 2x
y=ln 3x−22 x−64x 3−sin  5x9
1
1
y=e 3x  3x
e
y= x 3 −1ln x
y=cosh−1 cos x 3
2
4
4. A certain radioactive substance has half-life of 10 years. How long will it take for 50 grams
to decay to 4 grams? (Simplify answer as much as possible without calculator.)
5. Evaluate each integral.
(a)
arcsin 2x dx
(b)
(c)
∫
20x5
∫ 2x 2 x−7 dx
−5
∫ x x  ln x 2 dx
3
(d)
∫ 42x −7 dx
1
(e)
(f)
∫y
∫ 2cos x
0
1
(g)
arctan  y  dy

6
sin x dx
2
t 1
dt
∫ 2t 233t
−4
0
0
(h)
∫ 6 2x4 dx
−2
e 2x
(i) ∫ 2x
dx
e 5
5 x2
dx
(j) ∫
 1− x 6
x
dx
(k) ∫ 4
x 4
x3
(l) ∫ 4
dx
x 4
(m) ∫ 3x 4 x  dx
π
2 cos x
dx
(n) ∫
2
π 1+sin x
2
6. Find
 f −1 ' 5 given
7. Show that
f  x =
f  x =2 x 54x−1 .
sin x 1
is monotonic, i.e. that its inverse exists.
cos x
8. Find the area of the region bounded by
y=sinh x ,
y=0 and
x =ln 2 .
f (x )=6−tan−1( 2x)−5 ( x−1)3 has an inverse. (Explain your reasoning.)
Then, find ( f −1 )' (11)
9. Show that
10. Find the limits.
(a)
x →∞
(b)
( )
lim 1+
lim ( 1 )5x
x →∞
3
x
5x