The problem presented involves a piecewise function $f(t)$, where the integral is used to define $g(x) = \int_0^x f(t) \, dt$. Let's address the parts step by step:

(a) Evaluations of $g(x)$ for given values:

The integral evaluates the signed area under the graph $f(t)$. The given $g(x)$ values (for $x = 0, 3, 6, 9, 12, 15, 18$) have been correctly computed based on the graph:

• $g(0) = 0$
• $g(3) = 4.5$ (area of the triangle $1/2 \times 3 \times 3$)
• $g(6) = 0$ (positive and negative areas cancel out between $[0, 6]$)
• $g(9) = -4.5$ (area below the x-axis dominates)
• $g(12) = 0$ (symmetry of areas cancels out)
• $g(15) = 13.5$ (positive area from 12 to 15 added)
• $g(18) = 36$ (total positive area accumulated).

These are correct as checked.

(b) Estimate $g(21)$:

Using the midpoint rule for the interval $[18, 21]$, estimate the contribution from this interval and add it to $g(18)$:

1. The midpoint of $[18, 21]$ is $t = 19.5$, where $f(t) \approx 6$.
2. The width of the interval $\Delta t = 3$.
3. Approximate the area: $\Delta t \cdot f(19.5) = 3 \cdot 6 = 18$.
4. Add this to $g(18) = 36$: $g(21) \approx 36 + 18 = 54$.

The estimate in the solution incorrectly gives $g(21) = 66$, but this seems to overestimate the contribution from $[18, 21]$.

(c) Maximum and Minimum Values of $g(x)$:

1. Minimum: $g(x)$ reaches its minimum value at $x = 9$ with $g(9) = -4.5$, as this is the point where the most negative area is accumulated.
2. Maximum: $g(x)$ reaches its maximum at $x = 21$, with $g(21) \approx 54$ based on the corrected estimate.

Clarification and Observations:

• The integral $g(x)$ accumulates areas under $f(t)$, considering both positive and negative contributions.
• Overestimations in (b) likely stem from an overestimation of $f(t)$'s value at $t = 19.5$. This suggests careful use of the midpoint rule.

Would you like me to dive deeper into the calculation methods or the graph analysis? Here are related follow-up questions:

1. Can we use trapezoidal or Simpson's rule for a more accurate estimate of $g(21)$?
2. How does symmetry in $f(t)$ affect the integral's behavior for $g(x)$?
3. Can we derive a general formula for $g(x)$ for a piecewise-linear $f(t)$?
4. What would happen to $g(x)$ if $f(t)$ were shifted upwards or downwards?
5. Could we find where $g(x)$ is concave up or down using the second derivative?

Tip: When estimating areas, consider multiple rules (midpoint, trapezoidal, or Simpson's)