Problem 21 - Applied Calculus

Question

Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(f(x)=A e^{x}-B x^{2}+C\)

Step-by-Step Solution

Answer

The derivative is \(f'(x) = A e^{x} - 2Bx\).

Step 1: Differentiate the Exponential Term

To differentiate the term \(A e^{x}\), we apply the rule that the derivative of \(e^{x}\) is \(e^{x}\). Thus, the derivative of \(A e^{x}\) with respect to \(x\) is \(A e^{x}\).

Step 2: Differentiate the Power Term

For the term \(-Bx^{2}\), use the power rule, which states that \( \frac{d}{dx} x^{n} = n x^{n-1} \). In this case, \(-B\) is a constant multiplier, and we differentiate \(-Bx^{2}\) to get \(-2Bx\).

Step 3: Differentiate the Constant Term

The derivative of the constant term \(C\) is 0, as the derivative of any constant is 0.

Step 4: Combine All Derivatives

Combine the derivatives obtained for each term. The derivative of the function \(f(x) = A e^{x} - B x^{2} + C\) is \(f'(x) = A e^{x} - 2Bx + 0\). Simplifying, we find the derivative: \(f'(x) = A e^{x} - 2Bx\).

Key Concepts

Exponential Function

Exponential functions are fascinating in calculus due to their unique properties. An exponential function is often written as \( e^x \), where \( e \) is the base of natural logarithms, approximately equal to 2.71828. This function is unique because its derivative remains the same. This property makes exponential functions extremely useful in modeling growth processes, such as population growth, radioactive decay, and interest calculations.

• The derivative of \( e^x \) is \( e^x \).
• This holds true regardless of the coefficient in front of \( e^x \).

If we have a function like \( A e^x \), the derivative is simply \( A e^x \). Many real-world problems rely on solving differential equations involving exponential functions. They help us understand how quantities change over time smoothly and continuously.

Power Rule

In calculus, the power rule is a basic and essential tool for finding derivatives of polynomial functions. If you have a term \( x^n \), where \( n \) is a real number, the power rule tells you how to differentiate it quickly.

• The derivative of \( x^n \) is \( nx^{n-1} \).

For example, if you need to differentiate \( -Bx^2 \), apply the power rule: multiply by the exponent, 2, and decrease the power by one. So, the derivative becomes \( -2Bx \). This technique is straightforward and speeds up calculations, making it invaluable in various mathematical fields, from physics to economics.

Constant Function

A constant function is one that does not change, no matter the input. In mathematical terms, a constant function can be represented as \( f(x) = C \), where \( C \) is a constant number. In calculus, constant functions have a very simple derivative: zero. This is because derivatives measure change, and if a function is constant, it doesn’t change at all.

• The derivative of any constant is always 0.

In the function \( f(x) = A e^x - B x^2 + C \), the constant term \( C \) vanishes when differentiated. Knowing that the derivative of a constant is zero helps simplify functions and solve equations where constant terms are involved. Thus, understanding constant functions is key to mastering calculus and applying it effectively.