Problem 6 - Essential Calculus

Question

Find \(d y / d x\) and \(d^{2} y / d x^{2},\) and find the slope and concavity (if possible) at the given value of the parameter. $$ x=\sqrt{t}, y=3 t-1 \quad t=1 $$

Step-by-Step Solution

Answer

At \(t=1\), the slope of the curve is \(6\) and the curve is concave up (second derivative is \(6\)).

Step 1: Find dx/dt

\[x= \sqrt{t}\] is given. We have to differentiate it with respect to \(t\). The derivative is \[dx/dt = \frac{1}{2 \sqrt{t}}.\]

Step 2: Find dy/dt

\[y= 3t-1\] is given. Differentiating \(y\) with respect to \(t\), \[dy/dt = 3.\]

Step 3: Calculate dy/dx using Chain Rule

We calculate \(dy/dx\) using the chain rule: \[dy/dx = (dy/dt) / (dx/dt).\] Substituting the values that we got in step 1 and step 2, \[dy/dx = 3 / (1/(2\sqrt{t})).\] This simplifies to \[dy/dx = 6\sqrt{t}.\] At \(t=1\), \[dy/dx = 6.\]

Step 4: Calculate Second Derivative

We now find the second derivative \(d^{2} y / d x^{2}\) As the first derivative \((dy/dx)\) is dependent on t, we will differentiate it with respect to t and then divide the result by \(dx/dt\) (chain rule). \[\frac{d}{dt} (6 \sqrt{t}) = 3/t^{1/2},\] so \[d^{2} y/ d x^{2} = (3/t^{1/2}) / (1/(2\sqrt{t})) = 6\]. At \(t=1\), \[d^{2} y / d x^{2} = 6.\]

Step 5: Interpret Slope and Concavity

The sign of the first derivative tells us about the slope and that of the second derivative tells us about the concavity. A positive first derivative means that y increases as x increases, so the curve is rising at t = 1. A positive second derivative means the curve is concave up at t = 1.

Key Concepts

Chain Rule

The chain rule is an essential principle in calculus used to find the derivative of composite functions. When we have functions composed of other functions, like in parametric equations where both \(x\) and \(y\) are defined in terms of a third variable \(t\), the chain rule provides a way to differentiate these equations with respect to \(x\).

In the case of parametric differentiation, let's say we have \(x = f(t)\) and \(y = g(t)\), to find \(\frac{dy}{dx}\), we first find \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\), and then apply the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx} \]
This formula is based on the chain rule concept where \(dt\) in the numerator and denominator cancels out, leaving us with \(\frac{dy}{dx}\). The key is to treat \(dt\) like a common factor that cancels itself out. Use of the chain rule is crucial for working with parametric equations and understanding how rates of change in one variable affect another.

First Derivative

The first derivative of a function represents the rate at which the function's value changes with respect to change in the variable. In the given problem, finding \(\frac{dy}{dx}\) is determining the slope of the curve at any point defined by the parameter \(t\).

By taking the first derivative, we measure how the function \(y\) increases or decreases as \(x\) changes. In our exercise, once we obtained \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\), the first derivative \(\frac{dy}{dx}\) at any point \(t\) is found. When evaluated at \(t=1\), we get a specific slope, which in this case is 6. This number indicates that at the point where \(t=1\), for every increment in \(x\), \(y\) increases sixfold, giving the slope of the tangent to the curve at that point.

Second Derivative

The second derivative provides information about the concavity of a curve and the rate at which the slope changes. Taking the second derivative of a function, symbolically written as \(d^{2} y / d x^{2}\), helps us understand the curvature of the graph.

In the context of parametric equations, after finding first derivative \(dy/dx\) in terms of \(t\), we differentiate again with respect to \(t\) and then use the chain rule to divide by \(dx/dt\) to find the second derivative with respect to \(x\).
If \(d^{2} y / d x^{2} > 0\), the curve is concave up, indicating that as you move along the curve, the slope is increasing. In our problem, the second derivative is 6 at \(t=1\), implying a uniform concave up shape at this point on the curve.

Concavity of a Curve

Concavity refers to the direction in which a curve bends. A curve can be either concave up or concave down. Concavity is determined by the sign of the second derivative. If the second derivative is positive, the curve is concave up, which looks like a ‘U’. Conversely, if the second derivative is negative, the curve is concave down, resembling an inverted ‘U’.

In the exercise provided, because the second derivative is positive (6) at the given parameter \(t=1\), it tells us that the curve is concave up at that point. This means if you were to visualize the curve, it would bend upwards like a cup. Understanding concavity is essential for analyzing the behavior of graphs and for optimization problems in calculus where we need to find maximum and minimum points.