Homework 4

Directions:

  1. Show each step of your work and fully simplify each expression.
  2. Turn in your answers in class on a physical piece of paper.
  3. Staple multiple sheets together.
  4. Feel free to use Desmos for graphing.

Answer the following:

  1. Suppose a student sees a function \[f(x) = \dfrac{\sqrt{x}}{x^2 + x}\] and is asked to find $\dfrac{d}{dx} f(x)$. The student tries this: \[\dfrac{d}{dx} f(x) = \dfrac{d}{dx} \dfrac{\sqrt{x}}{x + 1} = \dfrac{\dfrac{d}{dx} \sqrt{x}}{\dfrac{d}{dx} (x^2 + x)}\] Why is this incorrect?
  2. Suppose a student tries again. They try \[\dfrac{d}{dx} f(x) = \dfrac{x^2 + x \cdot \frac{1}{2}x^{-1/2} - x^{1/2} \cdot 2x + 1}{(x^2 + x)^2}\] Why is this still incorrect?
  3. Given $g(r) = \sqrt{r} + \sqrt[3]{r}$, find the first and second derivatives.
  4. Suppose \[f(x) = 3\cdot(x^2 - 1)\] Do you need to use the product rule to find $f'(x)$? Why or why not?
  5. Find the derivative using the Differentiation Rules. Remember to observe your input variable and to fully simplify each expression.
    1. $y = \dfrac{x^2 + 1}{x^2 - 1}$
    2. $y = x + \dfrac{1}{x}$
    3. $y = \dfrac{1 + 2x}{3 - 4x}$
    4. $f(x) = \dfrac{x}{x + \frac{1}{x}}$
    5. $f(x) = (x^2 + x + 1)(x^{-2} + x^{-3})$
  6. The equation of motion of a particle is $s(t) = t^4 - 2t^3 + t^2 - t$ where $s$ is in meters and $t$ is in seconds.
    1. Find the velocity and acceleration as functions of $t$.
    2. Find the acceleration when the velocity is 0. (hint: set $v(t) = 0$ to find the time when the velocity is 0).
  7. Find an equation of the normal line to the curve $y = \sqrt{x}$ that is parallel to the line $2x + y = 1$.
    Hint:
    1. Find the derivative and set it equal to slope of the above line.
    2. Solve for $x$, this value of $x$ gives you the $x$-value of the point of tangency.
  8. Differentiate the following:
    1. $f(x) = \sec x$
    2. $f(x) = \cot x$
    3. $f(x) = x\cos x + 2\tan x$
    4. $f(x) = \dfrac{\sec x}{1 + \tan x}$
    5. $w(t) = \dfrac{\sin t}{1 - \sin t}$
  9. Find the 51st derivative of $f(x) = \sin x$.
  10. Suppose $F(x) = \sin(\cos (x))$. Finding a decomposition $f\circ g = F$, you determine that \[f(x) = x \qquad \qquad g(x) = \sin (\cos (x))\] Why is this decomposition useless when finding $F'(x)$?
  11. Differentiate the following. Be sure to recognize the form of the expression you are taking the derivative of in order to use the correct rule.
    1. $f(x) = (3x - x^3)^{23}$
    2. $f(x) = \sqrt[3]{1 + \tan x}$
    3. $f(x) = (3x + 2)^3(4x^2 + 3)^4$
    4. $f(x) = \dfrac{\sin^2(x)\cdot(4x^2-1)^3}{(3x^3 - 2x)^4}$
    5. $g(x) = \cos^3 x$
    6. $g(\theta) = \cos \theta^3$
    7. $y = \sqrt{x + \sqrt{ x + \sqrt{x}}}$
    8. $f(x) = \sqrt{\dfrac{x}{x^2 + 4}}$
    9. $f(x) = \sin x + \sin^2 x$
    10. $g(a) = \sin(a \sin a)$
    11. $f(x) = \sqrt{\sec x}$
    12. $f(x) = \left(\dfrac{\sin^2(x)}{(x^2 + 1)^3}\right)^4$
  12. Find the equation of the tangent line to the curve at the given point.
    1. $y = (1 + 2x)^{10}, \qquad (0, 1)$
    2. $y = \sin(\sin(x)), \qquad (\pi, 0)$

    13. The rest of these problems will appear on next week's homework. Skip the rest of these for this homework.

  13. What is the main idea behind implicit differentiation?
  14. What are the four steps to isolate an entity in an equation, let's say $\dfrac{dy}{dx}$?
  15. Isolate $z$ in the equation \[-\sin(xy)(x + xz) = 1 + \cos(y)z\]
  16. Find $dy/dx$ with implicit differentiation for the equation \[x^3 + y^3 + x^2 + y = 1\]
  17. Consider the equation $x^{2/3} + y^{2/3} = 4$. Find the equation of the tangent line at the point $(-3\sqrt{3}, 1)$.