Problem 69 - University Calculus: Early Transcendentals

Question

Find the value of \(a\) that makes the following function differentiable for all \(x\) -values.$$g(x)=\left\\{\begin{array}{ll} a x, & \text { if } x<0 \\ x^{2}-3 x, & \text { if } x \geq 0 \end{array}\right.$$

Step-by-Step Solution

Answer

The value of \(a\) is \(-3\).

Step 1: Understand Differentiability

To ensure a piecewise function is differentiable at a point, it must first be continuous at that point. Then, the derivatives on either side of the point must be equal.

Step 2: Check Continuity at x = 0

For the function to be continuous at \(x = 0\), the limit from the left \( \lim_{x \to 0^-} g(x) \) must equal the limit from the right \( \lim_{x \to 0^+} g(x) \) and also the value at \(x = 0\). Calculate both limits:- Left limit: \( \lim_{x \to 0^-} g(x) = \lim_{x \to 0^-} a x = 0 \).- Right limit: \( \lim_{x \to 0^+} g(x) = \lim_{x \to 0^+} (x^2 - 3x) = -0 = 0 \). Both limits give 0, so the function is continuous if \(g(0)\) is also 0, which is \(0^2 - 3 \times 0 = 0\). Thus, the function is continuous at \(x = 0\).

Step 3: Check Differentiability at x = 0

Calculate the derivatives of both parts of the function and ensure they are equal at \(x = 0\).- For \(x < 0\), the derivative \(g'(x) = a\).- For \(x \geq 0\), the derivative \(g'(x) = 2x - 3\).At \(x = 0\), set \(a\) equal to the derivative of the second part:\[a = 2 \times 0 - 3 = -3\].

Step 4: Verify Solution

Since both the continuity and differentiability conditions are satisfied with \(a = -3\), we conclude that \(a = -3\) makes the function differentiable for all \(x\).

Key Concepts

Piecewise Functions

Piecewise functions are mathematical expressions defined by multiple sub-functions, each with its own domain. The function exists as different expressions over various intervals of the input variable. They are typically used to outline small rules for different parts of their domain, often used to describe situations where a rule or pattern changes depending on the input value.
In this case, we have a piecewise function defined as:

• For values of \(x\) less than 0, the expression is \(ax\).
• For values of \(x\) greater than or equal to 0, the expression is \(x^2 - 3x\).

This setup visually represents the particular behavior of a function on distinct sub-domains of its input variable. Understanding piecewise functions involves knowing that each piece has its rule but applies only within a specific range of values.

Continuity

When we discuss continuity in piecewise functions, we want to know if there are any "gaps" or "jumps" in the function's graph. A function is continuous at a point if the limit from the left side of the point equals the limit from the right side, and also equals the function's value at that point. For instance, in our problem, we check continuity at \(x = 0\).
Here is what we need to consider:

• The limit of \(g(x)\) as \(x\) approaches 0 from the left must equal the limit of \(g(x)\) as \(x\) approaches 0 from the right.
• Additionally, the value of \(g(x)\) at \(x=0\) should equal both these limits.

Since both limits are 0 and the value at \(x=0\) in \(g(x) = x^2 - 3\times0 = 0\), the given function is continuous at this critical point. Hence, there's no abrupt change in function value at \(x=0\).

Derivatives

Derivatives in piecewise functions focus on the slope or rate of change for each sub-function in its domain segment. For a piecewise function to be differentiable, it first needs to be continuous at all points, especially at the junctions where the function's definition changes.
After ensuring continuity:

• Differentiate each part with respect to \(x\). Here, for the part \(ax\) when \(x<0\), the derivative \(g'(x)\) is \(a\).
• For \(x^2 - 3x\) when \(x \geq 0\), the derivative \(g'(x)\) is \(2x - 3\).

Next, examine whether these derivatives are equal at \(x=0\). The differential values should be equal for the function to be smooth without kinks or corners. Since they are equal when \(a = -3\), this ensures differentiability at this juncture.

Mathematics Problem Solving

Solving mathematics problems like this one requires a step-by-step approach, building upon foundational knowledge of piecewise functions, continuity, and derivatives. It involves:

• Understanding the condition for differentiability in piecewise functions.
• Checking that the function is continuous across sectional boundaries.
• Determining the derivative for each piece and matching them at the transition point.
• Validating the continuity and derivative conditions are honored by the solution.

Effective problem-solving combines analytical skills and patience, ensuring each criteria is satisfied before concluding. By setting \(a = -3\), as found above, the function meets all requirements for being differentiable and smooth for every value of \(x\).