MCV 4U – Calculus and Vectors
Applications of Derivatives
Winter 2016
TUTORBEE
Word Problems
Equation of the Tangent and Normal
1. Write the equation of a tangent to the curve:
𝑓 𝑥 =
1 % 1
𝑥 +
𝑥
𝑥
when 𝑥 = 1.
2. Find the equation of the normal line to the curve:
𝑥−2
𝑓 𝑥 =
)
𝑥
when 𝑥 = 1.
3. Find the equations of the tangent and normal to the curve:
𝑓 𝑥 =
)
𝑥 % − 2𝑥 + 4
%
at 𝑥 = 2.
4. Find the equation of the tangent to 𝑥 + − 6𝑥𝑦 − 2𝑦 + = 0 at (4,2)
5. Find the equation of the normal to:
𝑓 𝑥 =
𝑥% + 2
𝑥% − 2
when 𝑥 = 1.
6. Find the equation of the tangent to the function:
𝑦 = 𝑥 % − 8𝑥 + 14
that is parallel to the line:
𝑦 = 4𝑥 + 2𝑦 − 4
7. Show that the two curves 𝑦 = 2𝑥 + + 6𝑥 % + 6𝑥 + 3 and 𝑦 = 3𝑥 % + 6𝑥 + 3 touch each other. Find the
common tangent.
8. Show that there is no tangent to 𝑦 = 𝑥 % − 4𝑥 which passes through (2,1).
9. The graph of:
𝑓 𝑥 =
𝑎𝑥 + 𝑏
(𝑥 − 1)(𝑥 − 4)
has a horizontal tangent line at (2, −1). Find 𝑎 and 𝑏.
10. Determine the equations of both lines that are tangent to the graph of 𝑓 𝑥 = 𝑥 % and pass through
the point 1, −3 .
7
11. Show that the points of intersection of the quadratic functions 𝑦 = 𝑥 % and 𝑦 = − 𝑥 % , the tangents
to the functions are perpendicular.
%
12. Find the equation of the tangents to the curve: 25𝑥 % + 9𝑦 % = 225 that passes through (7,5).