Practise midterm 1.
Name:
CLOSED book, calculators allowed. Remember that points are given for the
steps that you perform, not just the answer.
1. (a) For f (x) = 7x5
4x2 + x
1, write down f 0 (x).
(b) For f (x) = tan x, nd f 0 (=2).
(c) For f (t) = (t2
(d) Compute
Z
3t2
4t) sin t, nd f 0 (t).
2 dt.
2. (a) For f (x) = sin4 (x2 + 3), nd f 0 (x).
(b) For f (x) = (x +
p
x2 + 1)7 , nd f 0 (x).
1
3. Use the chain rule to show that the derivative of an odd function is an even
function. [Recall that a function f (x) is odd if f ( x) = f (x), and is even
if f ( x) = f (x)].
4. Answer 'true' (T) or 'false' (F) by circling the appropriate letter.
T/F
\If the acceleration of an object is negative then its velocity
is negative."
T/F
\The function f (x) = 1 has only one antiderivative."
T/F
\Given a function f (x), the value f 0 (c) is dened precisely
when f (c) is dened."
T/F
\The derivative of a product is the product of the derivatives."
T/F
\If f (x) = 5 then f 0 (x) = 5 4 ."
5. Find the equation of the tangent line to the graph y = cot x at x =
2
.
4
6. A y is crawling from left to right along the top of the curve y = 7 x2 . A
spider waits at the point (4,0). Find the distance between the two insects
when they rst see each other.
3
7. Prove, using the limit denition of the derivative, that the derivative of cos x
is -sin x. Feel free to quote the following facts when you need them:
lim
!
h 0
sin h
cos h
= 1 and lim
h
!
0
h
h
4
1
=0