Yours

2.11: Homework- Instantaneous Velocity

Informal Calculus with Applications to Biological and Environmental Sciences · LibreTexts

0%

Contents

    Text

    Size
    100%
    Theme
    Typeface
    Line length
    Spacing

    From LibreTexts, under CC BY 4.0, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    1. The position of a falling object follows the equation \(f(t) = -16 t^2 + 64\) from \(t = 0\) to \(t = 2\).
      1. Verify that the points \((1, 48)\) and \((2, 0)\) are on the curve by computing \(f(1)\) and \(f(2)\) and verifying you get \(48\) and \(0\).
        This seems to work.
        ans
      2. Compute the slope of the line going through \((1, 48)\) and \((2, 0)\).
        -48
        ans
      3. Verify that the points \((1, 48)\) and \((1 + h, -16 h^2 - 32 h + 48)\) lie on the curve. You’ve already done \((1, 48)\), so now just simplify \(f(1 + h)\) and verify you get \(-16 h^2 - 32 h + 48\).
        This seems to work.
        ans
      4. Compute the slope of the line through \((1, 48)\) and \((1 + h, -16 h^2 - 32 h + 48)\) (hint: you should get \(-16h - 32\)!)
    2. The position of a falling object follows the equation \(f(t) = -5 t^2 + 45\) from \(t = 0\) to \(t = 3\).
      1. Verify that the points \((2, 25)\) and \((3, 0)\) line on the curve, and compute the slope through these two points.
        The slope is \(-25\)
        ans
      2. Verify that the points \((2, 25)\) and \((2 + h, -5 h^2 - 20h + 25)\) lie on the curve, and compute the slope of the line through these two points.
      The slope is \(-5h - 20\).
      ans
    3. Let \(g(t) = -10t^2 + 2000\) represent a population of goats, where \(g(t)\) is measured in goats and \(t\) is measured in years. Suppose \(t\) only works on the range from \(0\) to \(10\). This population is stabilizing during this period.
      1. Sketch a graph of \(g(t)\).
      2. Find the slope of the secant line hitting \(g(t)\) at \(t = 2\) and \(t = 3\).
        \(-50\) goats per year
        ans
      3. The slope of the secant line hitting \(g(t)\) at \(t = 2\) and \(t = 2+h\).
        \(-40-10h\) goats per year
        ans
      4. What is \(\lim_{h \to 0}\) for your answer in part (c)?
        \(-40\) goats per year.
        ans
      5. How quickly is the goat population growing at \(t = 2\)?
        \(-40\) goats per year.
        ans

    This page titled 2.11: Homework- Instantaneous Velocity was last modified on Tue, 19 Oct 2021 06:45:33 GMT and is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Tyler Seacrest via source content that was edited to the style and standards of the LibreTexts platform.