Problem 46 - Calculus - AP Edition

Question

Find (a) \(y^{\prime \prime \prime}(0),\) where \(y=4 x^{4}+2 x^{3}+3\) (b) \(\left.\frac{d^{4} y}{d x^{4}}\right|_{x=1},\) where \(y=\frac{6}{x^{4}}\)

Step-by-Step Solution

Answer

(a) 12; (b) 5040.

Step 1: Differentiate the function for part (a)

The given function is \( y = 4x^4 + 2x^3 + 3 \). To find the third derivative at \( x = 0 \), first, we differentiate it once to find \( y' \). It becomes \( y' = 16x^3 + 6x^2 \).

Step 2: Find the second derivative for part (a)

Differentiate \( y' = 16x^3 + 6x^2 \) with respect to \( x \) to get the second derivative \( y'' = 48x^2 + 12x \).

Step 3: Find the third derivative for part (a)

Differentiate \( y'' = 48x^2 + 12x \) to get the third derivative \( y''' = 96x + 12 \).

Step 4: Evaluate the third derivative at x = 0

Substitute \( x = 0 \) into \( y''' = 96x + 12 \) to find \( y'''(0) = 12 \).

Step 5: Differentiate the function for part (b)

The given function is \( y = \frac{6}{x^4} = 6x^{-4} \). Differentiate it once to find the first derivative \( y' = -24x^{-5} \).

Step 6: Continue derivatives to the fourth for part (b)

Differentiate the first derivative \( y' = -24x^{-5} \) to get \( y'' = 120x^{-6} \). Next, \( y''' = -720x^{-7} \). Finally, differentiate to get \( y^{(4)} = 5040x^{-8} \).

Step 7: Evaluate the fourth derivative at x = 1

Substitute \( x = 1 \) into \( y^{(4)} = 5040x^{-8} \) to find \( y^{(4)}(1) = 5040 \).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus. It involves finding the derivative of a function, which measures how the function's value changes as its input changes. Think of it as finding the slope of the function at a specific point. This slope indicates the rate of change. There are several steps to differentiate a function:

• Find the First Derivative: This involves differentiating the given function once. For instance, if you start with a polynomial like \( y = 4x^4 + 2x^3 + 3 \), you'll get the first derivative \( y' = 16x^3 + 6x^2 \).
• Find Higher Order Derivatives: These are derivatives of the derivative. The second derivative, for instance, is derived from the first derivative, and so on. For example, from \( y' = 16x^3 + 6x^2 \), the second derivative is \( y'' = 48x^2 + 12x \). Continuing this process gives the third derivative \( y''' = 96x + 12 \).
• Evaluate at Specific Points: Once you have the required derivative, you can substitute specific values, like \( x = 0 \), to find the derivative's value at that point. For \( y'''(x) \), evaluating at \( x = 0 \) provides \( y'''(0) = 12 \).

Through these steps, differentiation helps us understand the behavior of functions and their rate of change in detail.

Polynomial Functions

Polynomial functions are mathematical expressions involving terms with non-negative integer exponents of a variable. These functions are important because they are relatively easy to differentiate and analyze. A polynomial function has the form:\[y = a_n x^n + a_{n-1} x^{n-1} + \ ... + a_1 x + a_0\]Where each "\(a\)" is a coefficient and "\(n\)" indicates the highest degree.Differentiating polynomial functions is quite intuitive because of the power rule: if you have a term like \( x^n \), the derivative becomes \( nx^{n-1} \).

• Example: For \( y = 4x^4 + 2x^3 + 3 \), applying the power rule results in \( y' = 16x^3 + 6x^2 \).
• Polynomials are easy to handle since each differentiation reduces the power by one, eventually simplifying the function.

Understanding polynomial differentiation is crucial as it forms the base for more advanced calculus topics.

Rational Functions

Rational functions are ratios of two polynomial functions. They are expressed as \( y = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. A simple example of a rational function is \( y = \frac{6}{x^4} \), which can be rewritten as \( y = 6x^{-4} \).Differentiating rational functions can be more intricate than polynomials because they often require special rules like the quotient rule. However, in simpler forms, they can sometimes be treated with the power rule by rewriting the expression:

• Example: For \( y = \frac{6}{x^4} \), you rewrite it as \( y = 6x^{-4} \), and differentiate using the power rule to get \( y' = -24x^{-5} \).
• Continuing this process through higher-order derivatives yields \( y'' = 120x^{-6} \), \( y''' = -720x^{-7} \), and \( y^{(4)} = 5040x^{-8} \).
• These derivatives are useful for understanding the behavior of rational functions at specific points, like \( x = 1 \), where \( y^{(4)}(1) = 5040 \).

By mastering rational functions, you gain tools to handle a wide variety of problems involving fractions of polynomials.