Problem 60 - Calculus Single Variable

Question

Let \(f(x)=\tan (x) .\) For every \(t \in(-\pi / 2, \pi / 2),\) write down the equation of the tangent line to the graph of \(f\) at \((t, f(t))\) in the form $$ x=M(t)(y-\tan (t))+t $$ What function is \(M(t) ?\) What line results when you compute $$ x=\lim _{t \rightarrow(\pi / 2)^{-}}(M(t)(y-\tan (t))+t) ? $$ Is it an asymptote of \(f ?\)

Step-by-Step Solution

Answer

The function \( M(t) = \cos^2(t) \). The resulting line is \( x = \pi/2 \), which is a vertical asymptote of \( f(x) = \tan(x) \).

Step 1: Find the derivative of f(x)

The function is given as \( f(x) = \tan(x) \). To find the slope of the tangent line to the graph of \( f \) at \((t, f(t))\), we need to compute \( f'(x) \), the derivative of \( f \) with respect to \( x \). The derivative of \( \tan(x) \) is given by \( f'(x) = \sec^2(x) \).

Step 2: Evaluate the derivative at x = t

Now, substitute \( x = t \) into the derivative to find the slope of the tangent line at the point \( (t, f(t)) \). This gives us \( f'(t) = \sec^2(t) \). This means the slope of the tangent line at \( t \) is \( \sec^2(t) \).

Step 3: Find equation of the tangent line in given format

The general point-slope form of a line is \( y - f(t) = m(x - t) \), where \( m \) is the slope. Since \( x = M(t)(y - \tan(t)) + t \) is given, rearranging the typical point-slope form gives \( M(t) = \frac{1}{m} \). This means \( M(t) = \frac{1}{\sec^2(t)} = \cos^2(t) \). Thus, the equation becomes \( x = \cos^2(t)(y - \tan(t)) + t \).

Step 4: Evaluate the limit of x as t approaches pi/2 from the left

The task now is to evaluate the limit \( \lim_{t \to (\pi/2)^-}(M(t)(y - \tan(t)) + t) \) where \( M(t) = \cos^2(t) \). As \( t \to \pi/2 \) from the left, \( \cos(t) \to 0 \) and thus \( \cos^2(t) \to 0 \). Therefore, the limit becomes \( \lim_{t \to (\pi/2)^-}(0 \cdot (y - \tan(t)) + t) \), which simplifies to \( \pi/2 \).

Step 5: Determine if the resulting line is an asymptote

The resulting line from the limit is \( x = \pi/2 \). Since the function \( f(x) = \tan(x) \) approaches infinity as \( x \) approaches \( \pi/2 \) from the left, this line indeed acts as a vertical asymptote of the function \( f(x) \).

Key Concepts

Understanding Derivatives

Calculus often involves rates of change, and derivatives are the mathematical tools used to express this concept. Simply put, a derivative tells us how one quantity changes in response to change in another.
For functions like trigonometric functions, understanding their derivatives is crucial to analyzing their behavior.

• The derivative of a function at a point represents the slope of the tangent line to the function at that point.
• For the function \( f(x) = \tan(x) \), its derivative is \( f'(x) = \sec^2(x) \). This tells us how steeply the tangent rises or falls as \( x \) changes.
• When evaluating the derivative at a specific point, such as \( x = t \), we get \( f'(t) = \sec^2(t) \), which is the slope of the tangent line at that point \((t, f(t))\).

Understanding derivatives helps in finding the equation of tangent lines, a crucial step in understanding how functions behave locally.

Exploring Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are foundational to many areas of mathematics. They capture the relationship between angles and side lengths in right-angled triangles.

• The tangent function \( \tan(x) \) is the ratio of the sine and cosine functions, given by \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).
• The graph of \( \tan(x) \) is particularly interesting due to its repeating pattern and vertical asymptotes. Each period is \( \pi \), and the function spans all real numbers except where it has vertical asymptotes.
• The unique structure of trigonometric functions makes them fascinating yet complex, as they involve periodic behavior that can be both predictable and surprising.

These functions are used not just in theoretical mathematics but also in fields like engineering and physics, where wave patterns and oscillations often resemble trigonometric curves.

Vertical Asymptotes in Functions

A vertical asymptote is a line that a graph approaches but never touches or crosses, indicating that a function increases or decreases without bound. In the context of trigonometric functions, vertical asymptotes often occur when the denominator of a fraction approaches zero.

• For the tangent function, \( \tan(x) \), vertical asymptotes occur at \( x = \pi/2 + n\pi \) where \( n \) is an integer. This is because \( \tan(x) = \frac{\sin(x)}{\cos(x)} \) and the cosine of these values becomes zero.
• As \( x \) approaches these points from either side, \( \tan(x) \) goes towards positive or negative infinity. This is why the lines \( x = (2n+1)\pi/2 \) serve as vertical asymptotes for \( \tan(x) \).
• Recognizing vertical asymptotes helps in graphing functions accurately and understanding their limits and continuity.

By evaluating the limit \( \lim_{t \to (\pi/2)^-}\cos^2(t)(y - \tan(t)) + t \), it's clear that as \( t \) approaches \( \pi/2 \), the function approaches a vertical line, affirming the role of vertical asymptotes.