91Ó°ÊÓ

Evaluate the indicated limits by direct evaluation as in Examples \(10-14 .\) Change the form of the function where necessary. $$\lim _{p \rightarrow-1} \sqrt{p}(p+1.3)$$

Solve the given problems by finding the appropriate derivatives.What is the instantaneous rate of change of the first derivative of \(y\) with respect to \(x\) for \(y=(1-2 x)^{4}\) for \(x=1 ?\)

Evaluate the indicated limits by direct evaluation as in Examples \(10-14 .\) Change the form of the function where necessary. $$\lim _{x \rightarrow 1} \frac{\sqrt{x}-1}{x-1}$$

Solve the given problems by using implicit differentiation. A computer is programmed to draw the graph of the implicit function \(\left(x^{2}+y^{2}\right)^{3}=64 x^{2} y^{2}\) (see Fig. 23.45 and Example 7 on page 607 ). Find the slope of a line tangent to this curve at (2.00,0.56) and at (2.00,3.07)

Solve the given problems involving limits. A \(5-\Omega\) resistor and a variable resistor of resistance \(R\) are placed in parallel. The expression for the resulting resistance \(R_{T}\) is given by \(R_{T}=\frac{5 R}{5+R} .\) Determine the limiting value of \(R_{T}\) as \(R \rightarrow \infty\)

In Exercises \(65-72, \lim _{x \rightarrow a^{-}} f(x)\) means to find the limit as \(x\) approaches a from the left only, and \(\lim _{x \rightarrow a^{+}} f(x)\) means to find the limit as \(x\) approaches a from the right only. These are called one-sided limits. Solve the following problems. Is there a difference between \(\lim _{x \rightarrow 2^{-}} \frac{1}{x-2}\) and \(\lim _{x \rightarrow 2^{+}} \frac{1}{x-2} ?\)