91Ó°ÊÓ

Use the shortcut rules to mentally calculate the derivative of the given function. HINT [See Examples 1 and 2.] $$ f(x)=2 x^{4}+3 x^{3}-1 $$

Use the shortcut rules to mentally calculate the derivative of the given function. HINT [See Examples 1 and 2.] $$ f(x)=\frac{1}{x}+\frac{1}{x^{2}} $$

(from the GRE Economics Test) In a multiplant firm in which the different plants have different and continuous cost schedules, if costs of production for a given output level are to be minimized, which of the following is essential? (A) Marginal costs must equal marginal revenue. (B) Average variable costs must be the same in all plants. (C) Marginal costs must be the same in all plants. (D) Total costs must be the same in all plants. (E) Output per worker per hour must be the same in all plants.

(from the GRE economics test) A student has a fixed number of hours to devote to study and is certain of the relationship between hours of study and the final grade for each course. Grades are given on a numerical scale (0 to 100), and each course is counted equally in computing the grade average. In order to maximize his or her grade average, the student should allocate these hours to different courses so that (A) the grade in each course is the same. (B) the marginal product of an hour’s study (in terms of final grade) in each course is zero. (C) the marginal product of an hour’s study (in terms of final grade) in each course is equal, although not necessarily equal to zero. (D) the average product of an hour’s study (in terms of final grade) in each course is equal. (E) the number of hours spent in study for each course is equal.

What is a cost function? Carefully explain the difference between average cost and marginal cost in terms of (a) their mathematical definition, (b) graphs, and (c) interpretation.

Find the indicated derivative. $$ s=2.3+\frac{2.1}{t^{1.1}}-\frac{t^{0.6}}{2} ; \frac{d s}{d t} $$

In Exercises 59-64, find the equation of the tangent line to the graph of the given function at the point with the indicated \(x\) -coordinate. In each case, sketch the curve together with the appropriate tangent line. $$ f(x)=x^{3} ; x=-1 $$

Fuel Economy Your used Chevy's gas mileage (in miles per gallon) is given as a function \(M(x)\) of speed \(x\) in mph, where $$ M(x)=\frac{4,000}{x+3,025 x^{-1}} $$ Calculate \(M^{\prime}(x)\) and hence determine the sign of each of the following: \(M^{\prime}(40), M^{\prime}(55)\), and \(M^{\prime}(60)\). Interpret your results.

If a stone is dropped from a height of 400 feet, its height \(s\) after \(t\) seconds is given by \(s(t)=400-16 t^{2}\), with \(s\) in feet. a. Compute \(s^{\prime}(t)\) and hence find its velocity at times \(t=0\), \(1,2,3\), and 4 seconds. b. When does it reach the ground, and how fast is it traveling when it hits the ground? HINT [lt reaches the ground when \(s(t)=0 .\).]

Consider \(f(x)=x^{3}\) and \(g(x)=x^{3}+3\). How do the slopes of the tangent lines of \(f\) and \(g\) compare?