WRITTEN ASSIGNMENT 5
Hand in full solutions to the questions below. Make sure you justify all your work and include complete
arguments and explanations. Your asnwers must be clear and neatly written, as well as legible (no tiny
drawings or micro-handwriting please!). Your answers must be stapled, with your name and student number
at the top of each page.
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1. Problem: Show that the curve y = x2 intersects the curve y = 1/ x at a point where the tangent lines
to each curve are perpendicular to each other.
To solve this problem, follow these steps.
(a) Find the point P of intersection between the two curves.
(b) Find the slope of the tangent line to y = x2 at the point P .
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(c) Find the slope of the tangent line to y = 1/ x at the point P .
(d) Compare the two slopes and show that the two tangent lines are perpendicular to each other.
2. A tangent line is drawn to the hyperbola y = 1/x at a point P . Find the area of the triangle formed
by the tangent line and the coordinate axes and explain why the triangle always has the same area,
no matter where P is located on the hyperbola. Draw a sketch of the situation.
3. For each one of the following functions, compute the derivative in at least two different ways, for
example, if appropriate, by applying the power rule and by applying the product rule.
(a) y =
√
x4 −3x−1 +5 x
2
x
√
2
(b) f (t) = ( x − x )(5x3 + 32 x)
4. If f (x) = (x − a)(x − b)(x − c), show that
f 0 (x)
1
1
1
=
+
+
f (x)
x−a x−b x−c
where a, b, c are constants.
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