Problem 12 - Calculus for Biology and Medicine

Question

Find } c \text { so that } f^{\prime}(c)=0 . $$ $$ f(x)=(x+3)^{2} $$

Step-by-Step Solution

Answer

The value of \( c \) that makes \( f'(c) = 0 \) is \( c = -3 \).

Step 1: Differentiate the Function

To find where the derivative of the function is equal to zero, we need to first find the derivative of the function \( f(x) = (x+3)^2 \). We use the power rule \( \frac{d}{dx}[u^n] = nu^{n-1} \frac{du}{dx} \). Here, \( u = x + 3 \) and \( n = 2 \). The derivative \( f'(x) = 2(x+3)^{2-1}(1) = 2(x+3) \).

Step 2: Set the Derivative Equal to Zero

Next, we set the derivative \( f'(x) = 2(x+3) \) equal to zero to find the critical points. This gives us the equation \( 2(x+3) = 0 \).

Step 3: Solve for x

Solve the equation \( 2(x+3) = 0 \). First, divide both sides by 2 to get \( x + 3 = 0 \). Then, subtract 3 from both sides to solve for \( x \): \( x = -3 \).

Step 4: Conclusion

The value of \( c \) that makes the derivative zero is \( c = -3 \). This means \( f'(c) = 0 \) when \( c = -3 \).

Key Concepts

Derivative

In calculus, the derivative is a concept that measures how a function changes as its input changes. It's a fundamental idea for understanding rates of change and slopes of curves. Differentiation, the process of finding the derivative, tells us about the function's behavior - whether a function is increasing, decreasing, or steady at a point. In practical terms, the derivative can indicate the speed of a moving object or the growth rate of a population. To find the derivative of a function, such as our example function \( f(x) = (x+3)^2 \), we apply differentiation rules. One of these rules is the power rule, which simplifies finding derivatives for functions of the form \( u^n \). Understanding derivatives gives a mathematical way to explore patterns and changes in a wide array of scientific fields.

Power Rule

The power rule is a quick and handy tool for differentiating functions. This rule states: if you have a function \( u^n \), its derivative is \( n \, u^{n-1} \frac{du}{dx} \). Essentially, you bring down the exponent as a coefficient and reduce the exponent by one.Let's break this down with an example. For the function \( f(x) = (x+3)^2 \), we consider \( u = (x+3) \) and \( n = 2 \).

• First, the exponent 2 is brought down to multiply the function, resulting in \( 2(x+3)^{1} \).
• Second, the expression \( \frac{du}{dx} \) is simply the derivative of \( u \), which is 1, since the derivative of \( x+3 \) is 1.
• This gives us the derivative \( f'(x) = 2(x+3) \).

The power rule simplifies the process of finding derivatives, making it easier to handle functions with higher powers, especially in polynomials.

Critical Points

Critical points are important markers for understanding a function's behavior, particularly where changes happen, such as peaks, valleys, or turning points. To find critical points, we look for where the derivative equals zero or is undefined.In the problem we dealt with \( f'(x) = 2(x+3) \). Setting this equal to zero, \( 2(x+3) = 0 \), helps us find where the slope of the tangent line to the graph is horizontal, indicating a potential maximum, minimum, or saddle point:

• To solve for \( x \), divide both sides by 2, obtaining \( x+3 = 0 \).
• Subtract 3 from both sides to find \( x = -3 \).

Thus, \( x = -3 \) is the critical point for this function, indicating a significant change in the curve's behavior at this x-value. Understanding critical points helps in sketching graphs and analyzing function trends.