Geometrical interpretation of derivatives
Name
1. Interpret the function 𝐴(𝑥) = 𝑥 2 as the area of a square with dimensions 𝑥 by 𝑥.
a) Label the square with the dimensions.
b) Now assume each dimension is increased by a tiny amount 𝑑𝑥. Label each component of the
additional area.
c) Using the previous labeled drawing, write down an expression for the increase in area, 𝑑𝐴.
d) If 𝑑𝑥 were truly a tiny increase, which components of your expression for 𝑑𝐴 are dominant? Which
can ignored?
e) Solve your expression for 𝑑𝐴⁄𝑑𝑥 .
2. Repeat the steps in problem 1 for 𝐴(𝑥) = (𝑥 + 1)(𝑥 + 3).
3. Repeat the steps in problem 1 for the general function 𝐴(𝑥) = 𝑔(𝑥)𝑓(𝑥) to determine 𝑑𝐴⁄𝑑𝑥 . Begin
by assuming the dimensions are increased by 𝑑𝑓 and 𝑑𝑔. Once you have an expression for 𝑑𝐴, assume
𝑑𝑓
𝑑𝑔
that a tiny increase 𝑑𝑓is equal to 𝑑𝑥 𝑑𝑥 and that a tiny increase 𝑑𝑔 is equal to 𝑑𝑥 𝑑𝑥.
.
4. Repeat the steps in problem 1 for the 𝑉(𝑥) = 𝑥 3 to determine 𝑑𝑉⁄𝑑𝑥 . Interpret 𝑉(𝑥) as the volume
of a cube with dimensions of 𝑥 by 𝑥 by 𝑥.
5.
a) Draw a unit circle on an x-y coordinate plane. Draw an angle 𝜃 in the first quadrant in standard
position. If an angle is represented in radians, the angle is equal to length of the arc in the unit circle
intercepted by the angle. Label this arc 𝜃.
b) Construct a right triangle (Triangle 1) using the angle 𝜃 and the terminal side as a hypotenuse. Label
the point on the terminal side of the angle that crosses the unit circle using cos 𝜃 and sin 𝜃.
c) Imagine zooming in on the piece of the arc right next to the terminal side of the angle. Increase the
arc, 𝜃, by amount 𝑑𝜃. Assume that 𝑑𝜃 is small enough that in this region the arc can be assumed to be a
straight line. Also think about increasing the arc as moving the point on the terminal side by small
amount in the x direction, 𝑑 cos 𝜃, and y direction, 𝑑 sin 𝜃. Construct a right triangle (Triangle 2) using
𝑑𝜃, 𝑑 sin 𝜃, and 𝑑 cos 𝜃.
d) How does Triangle 2 compare to Triangle 1?
e) Use Triangle 2 to compute an expression for 𝑑 sin 𝜃⁄𝑑𝜃.
6.
a) Graph a function 𝑓(𝑥) = 𝑥 2 . Shade the area from 0 to 𝑥. Call this area 𝐴(𝑥).
b) Increase the shaded area by a tiny amount 𝑑𝑥. Write down an expression for 𝑑𝐴.
c). Solve your expression for 𝑑𝐴 for 𝑑𝐴⁄𝑑𝑥 .
d) What does your expression for 𝑑𝐴⁄𝑑𝑥 tell you about the function 𝐴(𝑥)?