4.1 The tangent line problem and slope

Tangent lines are crucial in calculus, representing the instantaneous direction of a curve at a specific point. They help us understand how functions change at any given moment, providing insight into rates of change and slopes.

Calculating tangent line slopes using limits is a fundamental skill in differential calculus. This process leads to the concept of derivatives, which allow us to find slopes at any point on a function without repeatedly using the limit definition.

The Tangent Line and Slope

Concept of tangent lines

• A tangent line touches a curve at a single point called the point of tangency without crossing the curve at that point
• Represents the instantaneous direction of the curve at the point of tangency
• The slope of the tangent line indicates the rate of change of the curve at the point of tangency
  • Positive slope means the curve is increasing at that point (uphill)
  • Negative slope means the curve is decreasing at that point (downhill)
  • Zero slope means the curve has a horizontal tangent line at that point (flat)

Slope calculation using limits

• The slope of a tangent line can be found using the limit definition of the derivative: $\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$, where $f(x)$ is the function and $x$ is the point of tangency
• To find the slope, evaluate the limit as $h$ approaches zero:
  1. Substitute $x+h$ for $x$ in the function
  2. Simplify the numerator by combining like terms and factoring out $h$
  3. Cancel $h$ from the numerator and denominator, if possible
  4. Evaluate the limit as $h$ approaches zero
• Example: Find the slope of the tangent line to $f(x) = x^2$ at $x = 3$
  1. $\lim_{h \to 0} \frac{f(3+h) - f(3)}{h} = \lim_{h \to 0} \frac{(3+h)^2 - 3^2}{h}$
  2. $\lim_{h \to 0} \frac{9+6h+h^2 - 9}{h} = \lim_{h \to 0} \frac{6h+h^2}{h}$
  3. $\lim_{h \to 0} \frac{h(6+h)}{h} = \lim_{h \to 0} (6+h)$
  4. $6+0 = 6$, so the slope of the tangent line at $x = 3$ is 6

Applications of tangent lines

• Instantaneous velocity is the speed of an object at a specific moment in time
  • Found by calculating the slope of the tangent line to the position function at that time
  • Example: If an object's position is given by $s(t) = t^2 + 2t$, the instantaneous velocity at $t = 3$ is the slope of the tangent line at that point
• Marginal cost is the change in total cost when producing one additional unit
  • Found by calculating the slope of the tangent line to the total cost function at a specific quantity
  • Example: If the total cost function is $C(x) = 100 + 5x + 0.2x^2$, the marginal cost at a production level of 30 units is the slope of the tangent line at $x = 30$

Tangent lines vs derivatives

• The derivative of a function at a point equals the slope of the tangent line to the function at that point: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$, where $f'(x)$ is the derivative of $f(x)$
• The derivative is a function that gives the slope of the tangent line at any point on the original function
  • Used to find the slope of the tangent line at any point without using the limit definition each time
  • Example: If $f(x) = x^2$, the derivative $f'(x) = 2x$ gives the slope of the tangent line at any point $x$ on the original function