91Ó°ÊÓ

Explain why \(f^{\prime}(x)\) could be positive or negative at a point where \(f(x) > 0\).

Explain why the slope of a secant line can be interpreted as an average rate of change.

Sketch the graph of \(f(x)=\ln |x|\) and explain how the graph shows that \(f^{\prime}(x)=\frac{1}{x}\).

State the Extended Power Rule for differentiating \(x^{n}\). For what values of \(n\) does the rule apply?

Expanding square The sides of a square increase in length at a rate of \(2 \mathrm{m} / \mathrm{s}\). a. At what rate is the area of the square changing when the sides are \(10 \mathrm{m}\) long? b. At what rate is the area of the square changing when the sides are 20 m long? c. Draw a graph that shows how the rate of change of the area varies with the side length.

Shrinking square The sides of a square decrease in length at a rate of \(1 \mathrm{m} / \mathrm{s}\) a. At what rate is the area of the square changing when the sides are \(5 \mathrm{m}\) long? b. At what rate are the lengths of the diagonals of the square changing?

Expanding isosceles triangle The legs of an isosceles right triangle increase in length at a rate of \(2 \mathrm{m} / \mathrm{s}\). a. At what rate is the area of the triangle changing when the legs are \(2 \mathrm{m}\) long? b. At what rate is the area of the triangle changing when the hypotenuse is 1 m long? c. At what rate is the length of the hypotenuse changing?

Use Theorem 3.11 to evaluate the following limits. $$\lim _{x \rightarrow 0} \frac{\sin 5 x}{3 x}$$

Shrinking isosceles triangle The hypotenuse of an isosceles right triangle decreases in length at a rate of \(4 \mathrm{m} / \mathrm{s}\). a. At what rate is the area of the triangle changing when the legs are \(5 \mathrm{m}\) long? b. At what rate are the lengths of the legs of the triangle changing? c. At what rate is the area of the triangle changing when the area is \(4 \mathrm{m}^{2} ?\)

Expanding circle The area of a circle increases at a rate of \(1 \mathrm{cm}^{2} / \mathrm{s}\) a. How fast is the radius changing when the radius is \(2 \mathrm{cm} ?\) b. How fast is the radius changing when the circumference is \(2 \mathrm{cm} ?\)