91Ó°ÊÓ

Solve the initial value problems. \begin{equation}\frac{d^{2} y}{d t^{2}}=1-e^{2 t}, \quad y(1)=-1 \quad \text { and } \quad y^{\prime}(1)=0 \end{equation}

Assume that \(f\) and \(g\) are differentiable functions that are inverses of one another so that \((g \circ f)(x)=x .\) Differentiate both sides of this equation with respect to \(x\) using the Chain Rule to express \((g \circ f)^{\prime}(x)\) as a product of derivatives of \(g\) and \(f .\) What do you find? (This is not a proof of Theorem 1 because we assume here the theorem's conclusion that \(g=f^{-1}\) is differentiable.)

Evaluate the integrals in Exercises \(47-70\) $$ \int \frac{3 d r}{\sqrt{1-4(r-1)^{2}}} $$

Find the derivative of \(y\) with respect to the given independent variable. \begin{equation}y=2^{x}\end{equation}

In Exercises \(57-70\) , use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$ y=\sqrt{\frac{t}{t+1}} $$

In Exercises \(57-70\) , use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$ y=\sqrt{\frac{1}{t(t+1)}} $$

Find the derivative of \(y\) with respect to the given independent variable. \begin{equation}y=3^{-x}\end{equation}

Evaluate the integrals in Exercises \(47-70\) $$ \int \frac{6 d r}{\sqrt{4-(r+1)^{2}}} $$

Find the limits in Exercises \(51-66\) $$ \lim _{x \rightarrow 0^{+}}\left(1+\frac{1}{x}\right)^{x} $$

Find the limits in Exercises \(51-66\) $$ \lim \left(\frac{x+2}{x-1}\right) $$