91Ó°ÊÓ

Which of the following statements is true? (A) If \(f(x)\) is continuous, then \(f(x)\) is differentiable. (B) If \(f(x)\) is differentiable, then \(f(x)\) is continuous.

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=e^{-x}\) at \(a=0\)

Differentiate $$ f(x)=a x^{3} $$ with respect to \(x\). Assume that \(a\) is a constant.

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=\ln (x+1) $$

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ f(x)=\sqrt[7]{x^{2}-2 x+1} $$

Find the derivative with respect to the independent variable. $$ f(x)=4 \cos ^{2} x+2 \cos x^{4} $$

Apply the product rule to find the normal line, in slope-intercept form, of \(y=f(x)\) at the specified point. \(f(x)=\frac{(2-x)(3-x)}{4}\), at \(x=-1\)

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=\ln (3 x+4) $$

Differentiate $$ f(x)=x^{3}+a $$ with respect to \(x\). Assume that \(a\) is a constant.

(a) Use the formal definition to find the derivative of \(y=\frac{1}{x}\) at \(x=2\). (b) Show that the point \(\left(2, \frac{1}{2}\right)\) is on the graph of \(y=\frac{1}{x}\), and find the equation of the normal line at the point \(\left(2, \frac{1}{2}\right)\). (c) Graph \(y=\frac{1}{x}\) and the tangent line at the point \(\left(2, \frac{1}{2}\right)\) in the same coordinate system.