Math 1522:
Review of Prerequisites
R1. Diagnostic Test A (Stewart, p xxvi): 1,2,4,8(e,g),10
R2. Sketch a graph of the following polynomials, solely using roots, symmetry, degree and sign
of the leading coefficient.
(a) f (x) = (x2 − 1)3
(b) f (x) = x3 − cx, c > 0
(c) f (t) = 10t − 1.86t2
(d) v(r) = r0 r2 − r3 , r0 > 0
(e) F (r) = GM r/R3 , G, M, R > 0
(f) F (R) = GM r/R, G, M, r > 0
R3. Evaluate the following limits:
e2x − 3e3x
x→∞ −4e2x + e3x
x
(d) lim −x
x→0 e
(a) lim
(b)
(e)
e2x − 3e3x
x→−∞ −4e2x + e3x
lim
(c)
lim
tan x
π
(f)
x→ 2
+
lim x ln x
x→0+
lim
x→∞
cos x
x
x−2
. Show all work, including the limiting
+x−6
behaviour from both sides for each asymptote. Graph the function.
R4. Find all vertical asymptotes for f (x) =
x2
R5. (a) State the defintion of continuity of f (x) at x = a
(b) State the defintion of the derivative of f (x) at x = a
(c) Use the definition to find f 0 (3) for f (x) = 1/x.
R6. Find the derivatives of the following functions.
3
(a) f (x) = sin(x )
(d) g(t) =
Z
(c) y =
2
1
(1 + t)2
x
x
(c) F (x) =
(a) y = x2 sin(πx)
(b) F (R) = GM r/R
Z
t
dt
1 + t3
Z
1
(b) f (t) =
1 + 2t2
(d) y =
0
sin t
dt.
t
x4
cos(t2 ) dt
1
R7. Evaluate the following definite and indefinite integrals.
Z
Z
Z
x3
x3
1 − x2
dx
(b)
dx
(c)
dx
(a)
x2
1 + x4
1 − x2
Z 9√
Z
Z π/8
u − 2u2
8
(d)
du
(e)
x(2x + 5) dx
(f)
sec(2θ) tan(2θ) dθ
u
1
0
Z 3
Z 1
Z 2
sin x
x
2
(g)
|x − 4| dx
(h)
dx
(i)
dx
2
1
+
x
(x
+
1)3
0
−1
1
Z
Z t
2
2
(j)
tan θ sec θ dθ
(k)
(t − s)2 ds
0
Z
R8. Evaluate (a)
1
2
1
d
dx 1 + x2
dx ,
d
(b)
dx
Z
1
2
1
dx ,
1 + x2
d
(c)
dx
Z
1
x
1
dt .
1 + t2
R9. (a) Write down the linear approximation of a function
f (x) at x = a.
√
(b)
Find
the
linear
approximation
for
f
(x)
=
1
+
x
at a = 0. Use it to approximate
√
√
1.1, 0.9.
√
(c) Find the linear approximation for f (x) = x at a = 1.
1