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Chapter 2: Problem 16
If \(f(x)=x^{2}+5\) and \(g(x)=x^{3}-7\), how much is \(f(-2) \cdot g(-2)\).
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A rectangle is required to have an area of 4 square feet but its dimensions may vary. If one side has length \(x\), express the perimeter \(p\) of the rectangle as a function of \(x\). We shall be working with functions constantly. Thus we may have to deal with (4) $$ s=t^{2} $$ (5) $$ y=x^{3}+3 x^{2}+5 $$
(a) Is the function \(\left(3 h^{2}+h\right) / h\) identical with the function \(3 h+1\) ? (b) Can we use the latter function in place of the former to determine the number approached by the former as \(h\) approaches 0 ? Justify your answer.
One arm of a right triangle is 3 units and the hypotenuse is \(x\) units. Write a formula for the length of the other arm. Suggestion: Use the Pythagorean theorem.
Find the derived functions of the following functions: (a) \(f(x)=2 x^{2}\). Ans. \(f^{\prime}(x)=4 x\). (b) \(y=-2 x^{2}\). (c) \(y=3 x\). Ans. \(y^{\prime}=3\). (d) \(d=-9 t\). (e) \(y=7\). Ans. \(d y / d x=0\) (f) \(s=6\).
State in words what the following represent symbolically: (a) \(\dot{s}=32 t\). (b) \(\dot{s}=5 t\). (c) \(\dot{v}=-32\). (d) \(y^{\prime}=4 x\). (e) \(\frac{d y}{d x}=8 x\). (f) \(f^{\prime}(x)=-3 x\).
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