91Ó°ÊÓ

Find the differential of the function at the indicated number. $$ f(x)=x^{4}-2 x^{3}+3 ; \quad x=0 $$

Differentiate the function. $$ g(x)=\ln \sqrt{\frac{x \cos x}{(2 x+1)^{3}}} $$

The weekly total cost in dollars incurred by the BMC Recording Company in manufacturing \(x\) compact discs is $$C(x)=4000+3 x-0.0001 x^{2} \quad 0 \leq x \leq 10,000$$ a. What is the actual cost incurred by the company in producing the 2001 st disc? The 3001 st disc? b. What is the marginal cost when \(x=2000\) ? When \(x=3000 ?\)

Marginal Average Cost of Producing Television Sets The Advance Visual Systems Corporation manufactures a 19 -inch LCD HDTV. The weekly total cost incurred by the company in manufacturing \(x\) sets is $$C(x)=0.000002 x^{3}-0.02 x^{2}+120 x+70,000$$ dollars. a. Find the average cost function \(\bar{C}(x)\) and the marginal average cost function \(C^{\prime}(x)\). b. Compute \(\bar{C}^{\prime}(5000)\) and \(\bar{C}^{\prime}(10,000)\), and interpret your results.

The air temperature at a height of \(h\) feet from the surface of the earth is \(T=f(h)\) degrees Fahrenheit. a. Give a physical interpretation of \(f^{\prime}(h)\). Give units. b. Generally speaking, what do you expect the sign of \(f^{\prime}(h)\) to be? c. If you know that \(f^{\prime}(1000)=-0.05\), estimate the change in the air temperature if the altitude changes from \(1000 \mathrm{ft}\) to \(1001 \mathrm{ft}\).

Range of an Artillery Shell The range of an artillery shell fired at an angle of \(\theta^{\circ}\) with the horizontal is $$ R=\frac{1}{32} \nu_{0}^{2} \sin 2 \theta $$ in feet, where \(v_{0}\) is the muzzle speed of the shell. Suppose that the muzzle speed of a shell is \(80 \mathrm{ft} / \mathrm{sec}\) and the shell is fired at an angle of \(29.5^{\circ}\) instead of the intended \(30^{\circ} .\) Estimate how far short of the target the shell will land.

Range of an Artillery Shell The range of an artillery shell fired at an angle of \(\theta^{\circ}\) with the horizontal is $$ R=\frac{1}{32} v_{0}^{2} \sin 2 \theta $$ in feet, where \(v_{0}\) is the muzzle speed of the shell. Suppose that the muzzle speed of a shell is \(80 \mathrm{ft} / \mathrm{sec}\) and the shell is fired at an angle of \(29.5^{\circ}\) instead of the intended \(30^{\circ}\). Estimate how far short of the target the shell will land.

Let \(g\) denote the inverse of the function \(f\). (a) Show that the point \((a, b)\) lies on the graph of \(f .\) (b) Find \(g^{\prime}(b)\) $$ f(x)=x^{5}+2 x^{3}+x-1 ; \quad(0,-1) $$

Continuous Compound Interest Formula See Section 3.5. Use l'Hôpital's Rule to derive the continuous compound interest formula $$ A=P e^{r t} $$ where \(A\) is the accumulated amount, \(P\) is the principal, \(t\) is the time in years, and \(r\) is the nominal interest rate per year compounded continuously, from the compound interest formula $$ A=P\left(1+\frac{r}{m}\right)^{m t} $$ where \(r\) is the nominal interest rate per year compounded \(m\) times per year.

Two curves are said to be orthogonal if their tangent lines are perpendicular at each point of intersection of the curves. In Exercises \(89-92\), show that the curves with the given equations are orthogonal. $$ x^{2}+2 y^{2}=6, \quad x^{2}=4 y $$