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Exercises - The Definition of the Derivative

1. Use the definition of the derivative to $f\,'(x)$, when $\displaystyle{f\,(x) = \frac{1}{2-3x}}$

2. Find $f\,'(x)$ using the definition of the derivative, where $\displaystyle{f\,(x)=\frac{1}{\sqrt{4-x}}}$, and then find the equation of the normal line at $x=3$

3. Find $f\,'(x)$ using the definition of the derivative, where $f\,(x)=x^2-4x+4$ and then determine where the tangent line to the graph of $y=f\,(x)$ is horizontal

4. Given $f\,'(x) = 3x^2-2x+1$

   1. Use the definition of the derivative to find $f\,'(x)$
   2. Find the equation of the tangent line to the graph of $y=f\,'(x)$ at $x=2$
   3. Find the point on the graph of $y=f\,'(x)$ where the tangent line is horizontal
   4. Find the equation of the normal line to the graph of $y=f\,'(x)$ at $x=-1$

5. A ball is thrown upward from the top of a building. The initial height is 640 ft, and the initial velocity (upward) is 64 ft/s. Its height above the ground is given by $h(t)=-16t^2+64t+640$. Use the definition of the derivative to answer the following questions:

   1. What is the instantaneous velocity at $t=1$ second?
   2. What is the height of the ball at $t=1$ second?
   3. When will the ball reach its maximum height?

6. The cost (in dollars) to make $x$ new computers is given by $C(x) = x^2+196x+400$. The expected revenue is given by $R(x)=-3x^2+660x$. Use the definition of the derivative to answer the following questions:

   1. How many units need to be made in order to maximize profits?
   2. What is the maximum profit that can be made?

7. Use the definition of the derivative to find the derivative of each function given

   1. $\displaystyle{f\,(x) = \frac{1}{\sqrt{1-2x}}}$

   2. $\displaystyle{h(x)=6-\sqrt{x+4}}$

   3. $\displaystyle{f(x)=\frac{1}{3}}$

   4. $\displaystyle{f(x)=\frac{3}{2x+1}}$

   5. $\displaystyle{g(x)=\frac{x}{x+1}}$

   6. $\displaystyle{h(x)=1+\sqrt{x}}$

8. For each function given below, find $f\,'(x)$ using the definition of the derivative, then find the equation of the tangent line to the graph of $y=f\,(x)$ at the given $x$-value.

   1. $\displaystyle{f\,(x) = \frac{1}{\sqrt{3x}}; \quad x=3}$

   2. $\displaystyle{f\,(x) = \frac{1}{1-2x}; \quad x=1}$

   3. $\displaystyle{f\,(x) = \frac{1}{x^2+1}; \quad x=-1}$

9. Given $\displaystyle{f\,(x) = \frac{2}{4-x}}$

   1. Find $f\,'(x)$ using the definition of the derivative
   2. Find the equation of the normal line to the graph of $y=f\,(x)$ at $x=2$.

10. Use the definition of the derivative to show that $f\,'(x) = \displaystyle{\frac{-1}{2x\sqrt{x}}}$ when $\displaystyle{f\,(x) = \frac{1}{\sqrt{x}}}$, and then find the equation of the normal line to the graph of the equation $y=f\,(x)$ at $x=4$

11. Given $\displaystyle{f\,(x) = \frac{1}{1+x}}$

    1. Use the definition of the derivative to find $f\,'(x)$
    2. Find the equation of the tangent line to the graph of $y=f(x)$ at $x=0$
    3. Find the equation of the normal line to the graph of $y=f(x)$ at $x=-2$