Chapter 4

Find the first and the second derivatives of each function. \(g(t)=t^{-5 / 2}-t^{1 / 2}\)

Differentiate $$ h(t)=\sqrt{a t}(t-a)+a t $$

Suppose that \(P(x)\) is a polynomial of degree \(4 .\) Is \(P^{\prime}(x)\) a polynomial as well? If yes, what is its degree?

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

Suppose that \(P(x)\) is a polynomial of degree \(k .\) Is \(P^{\prime}(x)\) a polynomial as well? If yes, what is its degree?

Find the first 10 derivatives of \(y=x^{5}\).

In Problems 84-87, assume that \(f(x)\) is differentiable. Find an expression for the derivative of \(y\) at \(x=2\), assuming that \(f(2)=\) \(-1\) and \(f^{\prime}(2)=1\). $$ y=\frac{f(x)}{x^{2}+1} $$

Find a second-degree polynomial \(p(x)=a x^{2}+b x+c\) with \(p(0)=3, p^{\prime}(0)=2\), and \(p^{\prime \prime}(0)=6\)

Assume that \(f(x)\) is differentiable. Find an expression for the derivative of \(y\) at \(x=2\), assuming that \(f(2)=\) \(-1\) and \(f^{\prime}(2)=1\). $$ =\frac{x^{2}+4 f(x)}{f(x)} $$

The position at time \(t\) of a particle that moves along a straight line is given by the function \(s(t) .\) The first derivative of \(s(t)\) is called the velocity, denoted by \(v(t) ;\) that is, the velocity is the rate of change of the position. The rate of change of the velocity is called acceleration, denoted by \(a(t) ;\) that is, $$ \frac{d}{d t} v(t)=a(t) $$ Given that \(v(t)=s^{\prime}(t)\), it follows that $$ \frac{d^{2}}{d t^{2}} s(t)=a(t) $$ Find the velocity and the acceleration at time \(t=1\) for the following position functions: (a) \(s(t)=t^{2}-3 t\) (b) \(s(t)=\sqrt{t^{2}+1}\) (c) \(s(t)=t^{4}-2 t\)

Assume that \(f(x)\) is differentiable. Find an expression for the derivative of \(y\) at \(x=2\), assuming that \(f(2)=\) \(-1\) and \(f^{\prime}(2)=1\). $$ y=[f(x)]^{2}-\frac{x}{f(x)} $$

Neglecting air resistance, the height \(h\) (in meters) of an object thrown vertically from the ground with initial velocity \(v_{0}\) is given by $$ h(t)=v_{0} t-\frac{1}{2} g t^{2} $$ where \(g=9.81 \mathrm{~m} / \mathrm{s}^{2}\) is the earth's gravitational constant and \(t\) is the time (in seconds) elapsed since the object was released. (a) Find the velocity and the acceleration of the object. (b) Find the time when the velocity is equal to \(0 .\) In which direction is the object traveling right before this time? in which direction right after this time?

Assume that \(f(x)\) is differentiable. Find an expression for the derivative of \(y\) at \(x=2\), assuming that \(f(2)=\) \(-1\) and \(f^{\prime}(2)=1\). $$ y=\frac{f(x)}{f(x)+x} $$

Assume that \(f(x)\) and \(g(x)\) are differentiable at \(x\). Find an expression for the derivative of \(y\) $$ y=\frac{f(x)}{[g(x)]^{2}} $$

Assume that \(f(x)\) and \(g(x)\) are differentiable at \(x\). Find an expression for the derivative of \(y\) $$ y=\frac{x^{2}}{f(x)-g(x)} $$

Assume that \(f(x)\) and \(g(x)\) are differentiable at \(x\). Find an expression for the derivative of \(y\) $$ y=\sqrt{x} f(x) g(x) $$

Assume that \(f(x)\) is a differentiable function. Find the derivative of the reciprocal function \(g(x)=1 / f(x)\) at those points \(x\) where \(f(x) \neq 0\)

Find the tangent line to the hyperbola \(y x=c\), where \(c\) is. a positive constant, at the point \(\left(x_{1}, y_{1}\right)\) with \(x_{1}>0 .\) Show that the tangent line intersects the \(x\) -axis at a point that does not depend on \(c\).

(Adapted from Roff, 1992) The males in the frog species Eleutherodactylus coqui (found in Puerto Rico) take care of their brood. On the other hand, while they protect the eggs, they cannot find other mates and therefore cannot increase their number of offspring. On the other hand, if they do not spend enough time with their brood, then the offspring might not survive. The proportion \(w(t)\) of offspring hatching per unit time is given as a function of (1) the probability \(f(t)\) of hatching if time \(t\) is spent brooding, and (2) the cost \(C\) associated with the time spent searching for other mates: $$ w(t)=\frac{f(t)}{C+t} $$