AP Calculus AB Study Notes

Continuous Function and Derivatives

• The continuous function $f$ is defined on the interval $[-5, 8]$.
• The graph of $f$ consists of four line segments.

Function $g$ Defined by Integration

• Define $g$ such that:

  • $g(0)$ and $g(-5)$ need to be evaluated.
  • Example parts follow:
  • (a)
    • Finding $g(0)$: Integral of $f$ from a to 0.
    • Finding $g(-5)$: Integral of $f$ from a to -5.

• (b)

  • Derivative $g'(x)$ Definition:
  • Given by:
    $g'(x) = 2 + f(x)$
  • Finding $g''(4)$ and $g''(-2)$:
    • For $g''(4)$: $g''(4) = f'(4) = -1$
    • For $g''(-2)$: $g''(-2)$ does not exist because $f$ is not differentiable.

Concavity and Behavior of $g$

• (c) Intervals where $g$ is concave down can be determined by examining where $g'(x)$ is decreasing:
  • Increasing $g'(x)$ means $g$ is concave up and vice versa.
  • From analysis, the graph of $g$ is concave down on intervals: (-2, 0) and (2, 8)

Composite Function $h$ Defined

• $h(x) = g(x^3 + 1)$
• To find $h'(1)$:
  • Use Chain Rule:
  • $h'(1) = g'(x^3+1) * (3x^2)$

Exponential Function $f(x)$

• Define $f(x) = e^{2x}$.

Tangents and Areas

• For the area in region R bounded by $f(x)$, compute:
  • (a) Write an equation for the tangent to $f$ at $x=1$
  • (b) Find the area of region R.
  • (c) Formulate the volume integral (do not evaluate).

Particle Motion and Calculations

• For a particle moving along x-axis where $v(t) = \dfrac{t^6 - 13t^4 + 12}{10t^3 + 3}$:
  • (a) Find acceleration at $t = ext{…}$
  • (b) Determine speed at certain times.
  • (c) Find position at time $t$ and examine movement towards origin.
  • (d) Analyze any time returning to initial position.

Area and Rate Calculations in Silo

• For the silo problem:
  • Use the table data to approximate via right Riemann or integrals.
  • Discussion of grain spoiling rates and unspoiled grain approximation.

Average Rates and Variables

• Average rate of change concerns for given functions.
  • Evaluate average and instantaneous rates of specific variables over specified intervals.

Considerations on Riemann Sums

• Apply left and right Riemann sums accordingly to approximations.