91Ó°ÊÓ

Calculate \(\frac{d y}{d x}\) using implicit differentiation. $$\sin y+2=x$$

Define the acceleration of an object moving in a straight line.

Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. $$\begin{array}{cccccc} x & -2 & -1 & 0 & 1 & 2 \\ \hline f(x) & 2 & 3 & 4 & 6 & 7 \\ f^{\prime}(x) & 1 & 1 / 2 & 2 & 3 / 2 & 1 \end{array}$$ $$\text { a. }\left(f^{-1}\right)^{\prime}(4) \quad \text { b. }\left(f^{-1}\right)^{\prime}(6) \quad \text { c. }\left(f^{-1}\right)^{\prime}(1) \quad \text { d. } f^{\prime}(1)$$

Given that \(f^{\prime}(3)=6\) and \(g^{\prime}(3)=-2,\) find \((f+g)^{\prime}(3)\)

Find the derivative the following ways: a. Using the Product Rule (Exercises 7-10 ) or the Quotient Rule (Exercises 11-14 ). Simplify your result. b. By expanding the product first (Exercises 7-10 ) or by simplifying the quotient first (Exercises 11-14 ). Verify that your answer agrees with part ( \(a\) ). $$f(x)=x(x-1)$$

If \(f\) is differentiable at \(a,\) must \(f\) be continuous at \(a ?\)

Where does sin \(x\) have a horizontal tangent line? Where does \(\cos x\) have a value of zero? Explain the connection between these two observations.

Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. $$\begin{array}{cccccc} x & -4 & -2 & 0 & 2 & 4 \\ \hline f(x) & 0 & 1 & 2 & 3 & 4 \\ f^{\prime}(x) & 5 & 4 & 3 & 2 & 1 \end{array}$$ $$\text { a. } f^{\prime}(f(0)) \quad \text { b. }\left(f^{-1}\right)^{\prime}(0) \quad \text { c. }\left(f^{-1}\right)^{\prime}(1) \quad \text { d. } \quad\left(f^{-1}\right)^{\prime}(f(4))$$

An object moving along a line has a constant negative acceleration. Describe the velocity of the object.

Find the derivative the following ways: a. Using the Product Rule (Exercises 7-10 ) or the Quotient Rule (Exercises 11-14 ). Simplify your result. b. By expanding the product first (Exercises \(7-10\) ) or by simplifying the quotient first (Exercises 11-14 ). Verify that your answer agrees with part ( \(a\) ). $$g(t)=(t+1)\left(t^{2}-t+1\right)$$