MATH 110, SECTION 003, WRITTEN ASSIGNMENT 4
Hand in full solutions to the questions below. Make sure to justify all your work, and include complete arguments and explanations. Your answers must be legible (no tiny drawings or micro-handwriting!). Your answers must be stapled, with your name and student number at the top of each page.
Questions marked with an asterisk are optional. Thinking about them will improve your understanding of the concepts we have covered, but will not count towards your grade on the assignment.
(1) Find the derivatives of each of the following functions.
(a) f ( x ) = sin( x cos( x )) .
(b) g ( x ) = p x cos( x ) + x
.
x + sin( x )
(c) h ( x ) = sin( x
2
) − cos( x
2
) .
(d) i ( x ) = e tan( x )
2 x +1 .
(2) Sketch each of the following functions
(a)
(b)
(c)
(d) f ( x ) = 2 x g ( x ) = | sin( x ) | h ( x ) = sin( | x | ) i ( x ) = e
− 3 x
.
(3) Find the equation of the tangent line to the curve y = cos( e x
) at x = ln( π/ 2).
(4) Let f ( x ) = sin( x 2 ). Use the intermediate value theorem to prove that there exists a point a such that the tangent line to the curve y = f ( x ) at ( a, f ( a )) passes through the point (5 , − 5).
Hint: first find an equation that a must solve, and then show that that equation has a solution using the intermediate value theorem.
(5 ∗ ) Use the chain rule and the sum rule to prove the product rule. That is, without using the product rule , show that whenever f and g are differentiable functions,
( f ( x ) g ( x ))
0
= f
0
( x ) g ( x ) + f ( x ) g
0
( x ) .
Hint: this is easy when f = g .
(6 ∗ ) Use the chain rule to prove that the derivative of ln( x ) is 1 /x .
1