Calculus 1 Assignment 4
Alex Cowan
[email protected]
Due Thursday, October 22nd at 5 pm
1. Differentiate the following functions with respect to x:
a) (ex +√x)3
x)
+q
b) log(x
2
√ x√x
c) xe
t sin x
d) 1+(sin
x)2
1
e) (tan x)− n
f) cos(arcsin
x)
√
g) 1 − x2
h) log(f (x)) (for arbitrary f )
2
i) ex
2
j) (ex )0
2
k) (ex )00
2. Redraw the following functions, and sketch their derivatives next to them:
3.
a) What is the equation of the line tangent to the function f (x) = log(x) at the point (1, 0)?
b) What is the equation of the line tangent to the function f (x) = x1 at the point (a, a1 )?
c) What is the equation of the line tangent to the function f (x) at the point (a, f (a)) (for arbitrary f and a)?
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4.
a) Show that if f is discontinuous at a, then it cannot be differentiable at a.
b) Give an example of a function f which is continuous at a point a but not differentiable at a.
5. Let
x2 sin
0
x sin
0
f (x) =
g(x) =
and
h(x) =
x2
0
1
x
1
x
if x 6= 0
if x = 0
if x 6= 0
if x = 0
if x is rational
if x is irrational
What are f 0 (0), g 0 (0), and h0 (0)?
6.
a) Prove that the derivative of an even function is odd.
b) Suppose f is even. What is f 0 (0)? Draw a picture that illustrates this phenomenon.
7. Alice was sick last week and missed the class where I introduced derivatives. Pretend you’re writing an
email to her that explains to her how a derivative can be viewed as a rate of change. She’s not allowed to ask
any follow up questions for some reason, so for her sake you should be clear and thorough!
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