Review for Exam 2 Spring 2016
1. For each of the following functions:
f (x + h) − f (x)
(a) Find limh→0
, (b) find the equation of the tangent line of at x = 0.
h
√
1
(i) f (x) = x + 1, (ii)f (x) =
, (iii) f (x) = x2 + x.
x+4
2. Differentiate and then simplify the following. Show all your work.
(a)
√
y = (x2 − 9x + 8) x + 1
(c)
y = ex
(e)
(g)
(i)
(k)
(m)
(o)
(q)
x2 − 1
x2 + 1
ex − e−x
= x
e + e−x
=p
sec(x2 + 1)
+ cos2 x = 1
x
= tan
2
x +1
ln x
=
x
= log2 (x2 + 1)
p
= ln
x2 + 1
(b)
y=
(d)
y
y = xe−x sin x
y = sin3 (πx + 1)
π
y = e−x sin
(x + 1)
6
(f)
(h)
y
y
(j)
y
y = ln(x2 − 4x + 7)
(l)
y
(n)
y
(p)
y
(r)
y = 2−x
2 −2x−7
2
2
y = x ln(x
+ 1)
x−1
y = ln
p x+1
y = ln (x2 + 1)
2
3.
Use implicit differentiation to find the lines that are (a) tangent and (b) normal to the curve at
the given point
(i)
y 2 x2
+
= 2 at (3, 2), (ii) x3 + y 3 − 4xy = 0 at (2, 2), (iii) x − sin(πxy) = 1 at (1, 1),
4
9
4. Sketch the derivative of the function given below.
2
1.5
1
1.0
0.5
1
-1
2
3
1
-1
2
3
-0.5
-1.0
-1
-1.5
-2.0
-2
5. Review all quizzes (≥ 7) and handouts.
You need to know the definition of the derivative, product rule, quotient rule, chain rule, implicit
differentiation, logarithmic differentiation, inverse trigonometric functions, and how to apply them.