Calculus Preview: Limits, Derivatives & Integrals

Big Questions in Calculus

• Calculus is introduced as the study that answers three overarching questions:
  • What is a function value getting close to?
    • Formal name: limit
    • Main topic of Chapter 2.
  • What is the slope (rate of change) at an exact point?
    • Formal name: derivative
    • Main topic of Chapter 3.
  • What is the area under a curve?
    • Formal name: integral
    • Reserved for second-term calculus (Calc 2).
• These three questions tie together the central themes of calculus: approaching values, instantaneous change, and accumulation.

Rate of Change & The Tangent Problem

• "Rate of change" is first illustrated with three straight lines:
  • $y = \frac23x - 1$
    • Rises 2, runs 3 (gentle uphill).
  • $y = -3x + 2$
    • Drops 3 every 1 unit (steep downhill).
  • $y = -1$
    • Horizontal line (zero slope).
• Key observations:
  • Straight lines have a constant rate of change (slope doesn’t vary).
  • Most real functions are not linear; their slopes vary from point to point.

Variable‐Slope Example: $y = x^2 + 2$

• Vertex at $(0,2)$, symmetric points $(\pm1,3)$, $(\pm3,11)$.
• Tangent line behavior:
  • At $x = 0$: slope $\approx 0$ (graph flattens).
  • At $x = 1$: slope $\approx 2$ (rises 2, runs 1).
  • At $x = -2$: slope $\approx -4$ (very steep downward).
• Shows how slope changes from negative to zero to positive as $x$ increases.

Tangent vs. Secant Lines

• Tangent line: touches the curve at exactly one point and has the same slope as the curve there.
• Secant line: passes through two points $(a,f(a))$ and $(b,f(b))$.
• Slope of a secant line:
  $m_{\text{sec}} = \frac{f(b) - f(a)}{b - a}$
• As $b \to a$ (or $a,b$ get "closer and closer"), the secant slope tends to the tangent slope.
  → Conceptual bridge to limits and derivatives.

Detailed Example 1: Estimating a Tangent Slope

• Function: $f(x)=2x^2-3$; Point of interest: $(2,5)$.
• Choose nearby $x$-values to play the role of $b$:
  • $2.1,\ 2.01,\ 2.001,\ 2.0001$ (progressively closer to 2).
• Calculator table of $f(x)$:
  • $f(2.1)=5.82$
  • $f(2.01)=5.0802$
  • $f(2.001)=5.008$
  • $f(2.0001)=5.0008$
• Compute slopes $m_{\text{sec}}$ to $(2,5)$:
  • $8.2,\ 8.02,\ 8.0,\ 8.0$ respectively.
• Conclusion: Tangent slope $m=8$ at $(2,5)$.
• Equation of tangent line (point-slope form): $y = 8\,(x - 2) + 5$
  • Recall general formula to memorize: $y = m\bigl(x - x<em>1\bigr) + y</em>1$.

Physics Connection: Velocity

• Average velocity over $[t<em>1,t</em>2]$ mirrors secant slope: $v<em>{\text{avg}} = \frac{h(t</em>2)-h(t<em>1)}{t</em>2-t_1}$
  • Same structure as $m_{\text{sec}}$.
• Instantaneous velocity at time $t$ = slope of tangent to the height–time graph.

Detailed Example 2: Falling Object

• Object dropped from a $144\text{-ft}$ cliff:
  $h(t)= -16t^2 + 144$
  (hits ground at $t=3\text{ s}$).
• Average velocity just after 2 s ($[2,2.01]$):
  • $h(2)=80$, $h(2.01)=79.3584$
  • $v_{\text{avg}} \approx \frac{79.3584-80}{2.01-2}= -64.2\ \text{ft/s}.$
• Average velocity just before 2 s ($[1.99,2]$):
  • $h(1.99)=80.6388$
  • $v_{\text{avg}} \approx \frac{80-80.6388}{2-1.99}= -63.8\ \text{ft/s}.$
• Instantaneous velocity at $t=2$ is estimated by midpoint of the two: $\approx -64\ \text{ft/s}$ (negative sign denotes downward direction).

Preview of Area & Integrals

• Goal: Find exact area under a curve → Integral.
• Example function: $y=(x-3)^2+1$; interval $x\in[0,3]$.
• Right-endpoint rectangles (width 1): heights $1,2,5$ ⇒ area $1+2+5=8$ (under-estimate).
• Left-endpoint rectangles: heights $10,5,2$ ⇒ area $10+5+2=17$ (over-estimate).
• Averaging the two approximations: $\tfrac{8+17}{2}=12.5$.
• True integral value (calculated in higher calculus) is $12$, illustrating that better methods / smaller rectangles give more accuracy.

Ethical & Practical Implications

• Instantaneous rates (derivatives) govern physics, engineering, economics, biology—any discipline needing real-time change data.
• Integral ideas (accumulated area) lead to total distance traveled, work done, probability, etc.
• Recognizing approximation vs. exactness encourages critical evaluation of numerical methods and error bounds.

Connections to Future Topics

• Chapter 2: Rigorous definition of the limit underlying both secant-to-tangent transition and average-to-instantaneous velocity.
• Chapter 3: Formal rules for computing derivatives without repeated secant approximation.
• Second-term Calculus: Integral calculus, Riemann sums → Fundamental Theorem of Calculus linking derivatives and integrals.

Key Formulas to Memorize

• Slope of a secant line: $m_{\text{sec}}=\dfrac{f(b)-f(a)}{b-a}$
• Point-slope form of a line: $y=m\bigl(x-x<em>1\bigr)+y</em>1$
• Average velocity (identical structure): $v<em>{\text{avg}}=\dfrac{h(t</em>2)-h(t<em>1)}{t</em>2-t_1}$

Takeaways

• Limits, derivatives, and integrals are three perspectives on “getting close,” “instantaneously changing,” and “accumulating.”
• Secant-to-tangent reasoning gives intuitive access to derivatives and instantaneous velocities.
• Rectangle (Riemann) approximations preview how integrals measure area; precision improves as rectangles narrow.
• Mastery of these foundational ideas enables deeper exploration of calculus in upcoming chapters and future courses.