91Ó°ÊÓ

The number of gallons \(N\) of regular unleaded gasoline sold by a gasoline station at a price of \(p\) dollars per gallon is given by \(N=f(p)\). (a) Describe the meaning of \(f^{\prime}(1.479)\). (b) Is \(f^{\prime}(1.479)\) usually positive or negative? Explain.

Consider the function \(f(x)=\sin \beta x\), where \(\beta\) is a constant. (a) Find the first-, second-, third-, and fourth-order derivatives of the function. (b) Verify that the function and its second derivative satisfy the equation \(f^{\prime \prime}(x)+\beta^{2} f(x)=0\) (c) Use the results in part (a) to write general rules for the even- and odd- order derivatives \(f^{(2 k)}(x)\) and \(f^{(2 k-1)}(x)\) [Hint: \((-1)^{k}\) is positive if \(k\) is even and negative if \(k\) is odd. \(]\)

This law states that the rate of change of the temperature of an object is proportional to the difference between the object's temperature \(T\) and the temperature \(T_{a}\) of the surrounding medium. Write an equation for this law. 109\. Find an equation of the parabola \(y=a x^{2}+b x+c\) that passes through \((0,1)\) and is tangent to the line \(y=x-1\) at \((1,0)\).

Find an equation of the parabola \(y=a x^{2}+b x+c\) that passes through \((0,1)\) and is tangent to the line \(y=x-1\) at \((1,0)\).

Let \(f\) be a differentiable function of period \(p\). (a) Is the function \(f^{\prime}\) periodic? Verify your answer. (b) Consider the function \(g(x)=f(2 x)\). Is the function \(g^{\prime}(x)\) periodic? Verify your answer.

Let \((a, b)\) be an arbitrary point on the graph of \(y=1 / x\), \(x>0 .\) Prove that the area of the triangle formed by the tangent line through \((a, b)\) and the coordinate axes is 2 .

Find the tangent line(s) to the curve \(y=x^{3}-9 x\) through the point \((1,-9)\)

(a) Find the derivative of the function \(g(x)=\sin ^{2} x+\cos ^{2} x\) in two ways. (b) For \(f(x)=\sec ^{2} x\) and \(g(x)=\tan ^{2} x\), show that \(f^{\prime}(x)=g^{\prime}(x)\).

(a) Show that the derivative of an odd function is even. That is, if \(f(-x)=-f(x)\), then \(f^{\prime}(-x)=f^{\prime}(x)\). (b) Show that the derivative of an even function is odd. That is, if \(f(-x)=f(x)\), then \(f^{\prime}(-x)=-f^{\prime}(x)\).

Find the equation(s) of the tangent line(s) to the parabola \(y=x^{2}\) through the given point. (a) \((0, a)\) (b) \((a, 0)\) Are there any restrictions on the constant \(a\) ?