How would you work out when this function turns and goes in a different direction? (like a turning point for a parabola):
s=0.2t(t-5.1)(t-9.1)
it will be halfway between the roots, or
turning point is when tis halfway between 5.1 and 9.1, or t=7.1. Put that into the s equation, and solve.
since it's a cubic, it's hard to do without calculus.
The easiest way is to examine its graph.
http://www.wolframalpha.com/input/?i=0.2t%28t-5.1%29%28t-9.1%29
Looks like t≈2.1 or 7.4
To determine when a function changes direction, such as a turning point for a parabola, you need to find the critical points of the function. In this case, we have the function:
s = 0.2t(t - 5.1)(t - 9.1)
To find the turning points of a function, we need to find the points where the derivative of the function is equal to zero. The first derivative gives us the rate of change of the function, and when it equals zero, it means the function is neither increasing nor decreasing.
Step 1: Find the first derivative of the function.
To do this, we'll apply the product rule and differentiate each term separately:
s = 0.2t(t - 5.1)(t - 9.1)
Applying the product rule:
ds/dt = 0.2(t - 5.1)(t - 9.1) + 0.2t[(t - 9.1) + (t - 5.1)]
Simplifying and combining like terms:
ds/dt = 0.2(t - 5.1)(t - 9.1) + 0.2t(2t - 14.2)
Step 2: Set the first derivative equal to zero and solve for t.
To find the points where the derivative equals zero, we set ds/dt = 0:
0 = 0.2(t - 5.1)(t - 9.1) + 0.2t(2t - 14.2)
Simplify and expand the equation:
0 = 0.2(t^2 - 14.2t + 45.41) + 0.2(2t^2 - 14.2t)
0 = 0.2t^2 - 2.84t + 9.082 + 0.4t^2 - 2.84t
Combine like terms:
0.6t^2 - 5.68t + 9.082 = 0
Step 3: Solve the quadratic equation for t.
We now have a quadratic equation. We can factor it or use the quadratic formula to find the roots. Factoring might not be straightforward, so let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 0.6, b = -5.68, and c = 9.082. Plugging in these values to the formula:
t = (-(-5.68) ± √((-5.68)^2 - 4 * 0.6 * 9.082)) / (2 * 0.6)
Simplifying:
t = (5.68 ± √(32.3344 - 21.8136)) / 1.2
t = (5.68 ± √10.5208) / 1.2
We get two possible solutions for t:
t ≈ 1.897 or t ≈ 7.43
These are the values of t at which the function may change direction or have a turning point. To confirm if these points are indeed turning points, we can analyze the concavity of the function by finding the second derivative and check its sign at these points.