Mathematical Scratchpad

Math 151 Review for final.

  1. If the graph of $f$ is given together with its first and second derivatives below. Label the graphs for $f,f^{\prime }$ and $f^{\prime \prime }$
    graphics/finalreview__4.png

    Ans: $f$ is red, $f^{\prime }$is thin and $f^{\prime \prime }$is black thick.

  2. If the graph of $f$ is given below: MATH

    1. Compare $f$ $^{\prime }(0),f$ $^{\prime }(1),f$ $^{\prime }(2),f$ $^{\prime }(5)$ in increasing order. [ Ans: MATH

    2. Suppose the function $f$ (graph of $y=f(t)$, is given above) represent the distance that a car travels.

      1. When do you think the car has zero velocity between $t=0$ and $t=6$? [Ans: look for the point where $f^{\prime }(x)=0,$so it is about  x=4 (max of f) or x=0.8 (min of  f ].

      2. Approximate the time when the car reaches its highest speed for $t=0$ and $t=4$. [Note: speed = MATH].

        [Ans: We look for the slope of MATH to be the largest, so it is the inflection point of $f$.]

  3. Find the tangent line equations and the $y-intercepts$ at a given point for the following functions:

    1. $f(x)=\sqrt{3x+2}$ at $x=1.$[Ans: y=0.6708 x+1.565]

    2. $f(x)=x^{2}+x.$ at $x=-2.$[y=-3*x-4]

  4. Find MATH[Hint: Identify a function $f$ and a point $a$ and use the derivative at one point.]

    [Ans. This limit is the same as $f^{\prime }(a)$ when we pick $f(x)=\log (x)$ and $a=2,$so it is MATH ]

  5. Find the following limits with explanations:

    1. MATH[ans. MATH]

    2. MATH[$=\allowbreak 3.0$

    3. MATH[$=\allowbreak 3.0$

    4. MATH[=0

    5. MATH [=0

    6. MATH [=0

  6. Let the graph of a velocity function (x = time, y = feet/sec) be given below: Assuming the x-intercepts for the following graph is at $x=1,3,$ and $5.$ MATH

    1. What is the initial velocity (when time=0)? [Ans: Since when $x=0$, $y $ is not found, so we do not have enough info to find the initial velocity.]

    2. Explain how velocity function can be negative sometimes. [Ans. When we travel in opposite direction, the velocity is negative.]

    3. Estimate the inflection point(s) for the distance function$.$[Ans, they are at the local min and the local max of the velocity, so it about $x=2 $ and $x=4$].

    4. Estimate the maximum and minimum for the distance function$.$[Ans, max is at $x=1,5$ and minimum at $x=3.]$

    5. Find the interval(s) where the acceleration is negative. [Ans. This is where the velocity has negative slopes].

  7. If the graphs of MATH and MATH are given below: Then

    1. identify the graph for MATH and MATH respectively. [Ans. $y=f(x)=thick,$ $y=f^{\prime }(x)$ is the one always positive and the last one is for MATH ]

    2. find the interval(s) where $f$ is increasing or decreasing, [ans. $f$ is always increasing]

    3. find the maximum and minimum for $f$ in the interval $[0,2\pi ],$[since $f$ is increasing in $[0,2\pi ],$ the minimum is at $x=0$ and the maximum is at $x=2\pi .]$

    4. find the interval(s) where $f$ is concave upward and concave downward. [Since MATH in $(-3.14,0)$ , $f$ is concave in this interval;similarly, $f^{\prime }<0$ in $(0,3.14),$ $f$ is concave down this interval.]

      MATH

  8. Find the first derivative for the following functions:

    1. MATH[ MATH

    2. MATH [ MATH

    3. MATH[ MATH

    4. MATH[done in class]

    5. $f(x)=3^{x}x^{3},$ [ MATH

    6. MATH[ MATH

  9. If the derivative of a function is $\dfrac{x+1}{x-3}.$Then

    1. find the interval(s) where original function $f$ is increasing and decreasing, [hint: draw MATH

    2. find the interval(s) where the original function $f$ is concave upward and concave downward. [hint: find MATH and use the signs of $f^{\prime \prime }$ to find the intervals where $f$ is concave upward or downward]

    3. graph one possible function $f$ which has the derivate function $(x+1)(x-3).[$**this is a typo, ignore this one]

  10. Find $\dfrac{dy}{dx}$ if MATH[Use implicit differentiation to find $y^{\prime }:$ MATH

  11. Find $\dfrac{dy}{dx}$ if MATH [ MATH].

  12. Find MATH[MATH

  13. Find the first derivative for the following functions:

    1. MATH[ MATH

    2. MATH[ MATH

  14. Use the product or quotient rule to find MATH

  15. Find the followings:

    1. MATH [MATH

    2. MATH [MATH

  16. If $f(x)=e^{2x}\cos x.$ Find $f^{\prime }(x)$ by hand.

  17. Suppose the cost, in dollars, for a company to produce $x$ pairs of a new line of jeans isMATH

    1. Find the marginal cost function. [ MATH

    2. Find $C^{\prime }(100)$ and explain its meaning. What does it predict? [This will predict the cost for the 101st unit]

    3. Estimate the cost of making the 99 th pair of jeans. [Use $C^{\prime }(98)=$ use calculator to find this value]

  18. If MATH
    graphics/finalreview__131.png

    1. use the signs of $f$ $^{\prime }$to find the intervals where $f$ is increasing or decreasing, [hint: plot MATH, which is shown above. We need find the zero of MATHwhich is at $x=1$ so $f$ is increasing in $(1,\infty )$ and $f$ is decreasing in $(-\infty ,1).$ ]

    2. find the relative maximum and minimum for $f.$ [ $f$ has a relative minimum at $x=1]$

    3. find MATH[hint: MATH

    4. use the signs of $f^{\prime \prime }$ to find the intervals where $f$ is concave upward or downward. [hint: after finding MATH you need to plot MATH