Problem 1
Find the stationary values of the following functions: \((a) y=x^{3}\) (b) \(y=-x^{4}\) (c) \(y=x^{6}+5\) Determine by the Nth-derivative test whether they represent relative maxima, relative minima, or inflection points.
Problem 1
Find the value of the following factorial expressions: \((a) 5 !\) (b) \(8 !\) (c) \(\frac{4 !}{3 !}\) (d) \(\frac{6 !}{4 !}\) \((e) \frac{(n+2) !}{n !}\)
Problem 1
Find the relative maxima and minima of \(y\) by the second-derivative test: (a) \(y=-2 x^{2}+8 x+25\) (b) \(y=x^{3}+6 x^{2}+9\) (c) \(y=\frac{1}{3} x^{3}-3 x^{2}+5 x+3\) (d) \(y=\frac{2 x}{1-2 x} \quad\left(x \neq \frac{1}{2}\right)\)
Problem 1
Find the stationary values of the following (check whether they are relative maxima or minima or inflection points), assuming the domain to be the set of all real numbers: $$(a) y=-2 x^{2}+8 x+7$$ $$(b) y=5 x^{2}-x$$ $$(c) y=3 x^{2}+3$$ $$(d) y=3 x^{2}-6 x+2$$
Problem 2
Find the stationary values of the following functions: \((a) y=(x-1)^{3}+16\) (c) \(y=(3-x)^{6}+7\) (b) \(y=(x-2)^{4}\) \((d) y=(5-2 x)^{4}+8\) Use the Nth-derivative test to determine the exact nature of these stationary values.
Problem 2
Which of the following quadratic functions are strictly convex? (a) \(y=9 x^{2}-4 x+8\) (c) \(u=9-2 x^{2}\) (b) \(w=-3 x^{2}+39\) \((d) v=8-5 x+x^{2}\)
Problem 2
Find the first five terms of the Maclaurin series (i.e., choose \(n=4\) and let \(x_{0}=0\) ) for: (a) \(\phi(x)=\frac{1}{1-x}\) (b) \(\phi(x)=\frac{1-x}{1+x}\)
Problem 2
Find the stationary values of the following (check whether they are relative maxima or minima or inflection points), assuming the domain to be the interval \((0, \infty)\) $$(a) y=x^{3}-3 x+5$$ $$(b) y=\frac{1}{3} x^{3}-x^{2}+x+10$$ $$(c) y=-x^{3}+4.5 x^{2}-6 x+6$$
Problem 2
Mr. Creenthumb wishes to mark out a rectangular flower bed, using a wall of his house as one side of the rectangle. The other three sides are to be marked by wire netting, of which he has only 64 ft available. What are the length \(L\) and width \(W\) of the rectangle that would give him the largest possible planting area? How do you make sure that your answer gives the largest, not the smallest area?
Problem 3
A firm has the following total-cost and demand functions: \(C=\frac{1}{3} Q^{3}-7 Q^{2}+111 Q+50\) \(Q=100-P\) (a) Does the total-cost function satisfy the coefficient restrictions of (9.5)\(?\) (b) Write out the total-revenue function \(R\) in terms of \(Q\) (c) Formulate the total-profit function \(\pi\) in terms of \(Q\) (d) Find the profit-maximizing level of output \(Q^{*}\) (e) What is the maximum profit?
