Problem 40
Find the equations of the tangent and normal lines to the graph of \(g\) at the indicated point. \(g(\theta)=\frac{\cos \theta-8 \theta}{\theta+1}\) at (0,1).
Problem 41
Compute \(\frac{d}{d x}(\ln (k x))\) two ways: (a) Using the Chain Rule, and (b) by first using the logarithm rule \(\ln (a b)=\ln a+\ln b\), then taking the derivative.
Problem 41
Find the \(x\) -values where the graph of the function has a horizontal tangent line. $$f(x)=6 x^{2}-18 x-24$$
Problem 41
Use logarithmic differentiation to find \(\frac{d y}{d x}\), then find the equation of the tangent line at the indicated \(x\) -value. $$y=\frac{x+1}{x+2}, \quad x=1$$
Problem 42
Find the \(x\) -values where the graph of the function has a horizontal tangent line. \(f(x)=x \sin x\) on [-1,1].
Problem 42
Compute \(\frac{d}{d x}\left(\ln \left(x^{k}\right)\right)\) two ways: (a) Using the Chain Rule, and (b) by first using the logarithm rule \(\ln \left(a^{p}\right)=p \ln a,\) then taking the derivative.
Problem 42
Use logarithmic differentiation to find \(\frac{d y}{d x}\), then find the equation of the tangent line at the indicated \(x\) -value. $$y=\frac{(x+1)(x+2)}{(x+3)(x+4)}, \quad x=0$$
Problem 43
Find the \(x\) -values where the graph of the function has a horizontal tangent line. $$f(x)=\frac{x}{x+1}$$
Problem 43
The "wind chill factor" is a measurement of how cold it "feels" during cold, windy weather. Let \(W(w)\) be the wind chill factor, in degrees Fahrenheit, when it is \(25^{\circ} \mathrm{F}\) outside with a wind of \(w\) mph. (a) What are the units of \(W^{\prime}(w)\) ? (b) What would you expect the sign of \(W^{\prime}(10)\) to be?
Problem 44
Find the derivatives of the following functions. (a) \(f(x)=x^{2} e^{x} \cot x\) (b) \(g(x)=2^{x} 3^{x} 4^{x}\)
