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