Chapter 3: Problem 7
In Exercises \(7-14,\) find the differential \(d y\) of the given function. $$ y=3 x^{2}-4 $$
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In Exercises 61 and 62, use a graphing utility to graph the function. Then graph the linear and quadratic approximations \(P_{1}(x)=f(a)+f^{\prime}(a)(x-a)\) and \(P_{2}(x)=f(a)+f^{\prime}(a)(x-a)+\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}\) in the same viewing window. Compare the values of \(f, P_{1},\) and \(P_{2}\) and their first derivatives at \(x=a .\) How do the approximations change as you move farther away from \(x=a\) ? \(\begin{array}{ll}\text { Function } & \frac{\text { Value of } a}{a} \\\ f(x)=2(\sin x+\cos x) & a=\frac{\pi}{4}\end{array}\)
Find \(a, b, c,\) and \(d\) such that the cubic \(f(x)=a x^{3}+b x^{2}+c x+d\) satisfies the given conditions. Relative maximum: (2,4) Relative minimum: (4,2) Inflection point: (3,3)
Let \(f\) and \(g\) represent differentiable functions such that \(f^{\prime \prime} \neq 0\) and \(g^{\prime \prime} \neq 0\). Prove that if \(f\) and \(g\) are positive, increasing, and concave upward on the interval \((a, b),\) then \(f g\) is also concave upward on \((a, b)\).
Assume that \(f\) is differentiable for all \(x\). The signs of \(f^{\prime}\) are as follows. \(f^{\prime}(x)>0\) on \((-\infty,-4)\) \(f^{\prime}(x)<0\) on (-4,6) \(f^{\prime}(x)>0\) on \((6, \infty)\) Supply the appropriate inequality for the indicated value of \(c\). $$ g(x)=3 f(x)-3 \quad g^{\prime}(-5) \quad 0 $$
Engine Efficiency The efficiency of an internal combustion engine is Efficiency \((\%)=100\left[1-\frac{1}{\left(v_{1} / v_{2}\right)^{c}}\right]\) where \(v_{1} / v_{2}\) is the ratio of the uncompressed gas to the compressed gas and \(c\) is a positive constant dependent on the engine design. Find the limit of the efficiency as the compression ratio approaches infinity.
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