91Ó°ÊÓ

Radioactive Decay Suppose \(W(t)\) denotes the amount of a radioactive material left after time \(t\). Assume that \(W(0)=6\) and that $$ \frac{d W}{d t}=-3 W(t) $$ (a) How much material is left at time \(t=4 ?\) (b) What is the half-life of the material?

Find the derivative of $$ f(x)=\tan ^{3}\left(3 x^{3}-3\right) $$

Use logarithmic differentiation to find the first derivative of the given functions. $$ y=x^{x^{x}} $$

Radioactive Decay Suppose \(W(t)\) denotes the amount of a radioactive material left after time \(t .\) Assume that \(W(0)=10\) and \(W(1)=8\) (a) Find the differential equation that describes this situation. (b) How much material is left at time \(t=5 ?\) (c) What is the half-life of the material?

Use logarithmic differentiation to find the first derivative of the given functions. $$ y=\left(x^{x}\right)^{x} $$

Find the tangent line, in slope-intercept form, of \(y=f(x)\) at the specified point. $$ f(x)=\frac{3}{x}-\frac{4}{\sqrt{x}}+\frac{2}{x^{2}}, \text { at } x=1 $$

Allometric equations describe the scaling relationship between two measurements, such as skull length versus body length. In vertebrates, we typically find that [skull length] \(\propto\) [body length] \(^{a}\) for \(0

Find the derivative of $$ f(x)=\sec ^{2}\left(2 x^{2}-2\right) $$

Find the first and the second derivatives of each function. \(f(x)=x^{3}-3 x^{2}+1\)

Use logarithmic differentiation to find the first derivative of the given functions. $$ y=x^{\cos x} $$