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find a value of $k,$ if any, making $h(x)$ continuous on [0,5] $h(x)=\left\{\begin{array}{ll}k \cos x & 0 \leq x \leq \pi \\ 12-x & \pi<x\end{array}\right.$

Name: Problem 49, Chapter 1: Calculus Single Variable
Uploaded: 2021-05-18T04:53:19-08:00
Duration: 3 min 30 s
Channel: Dwijendra Rao
Description: find a value of k, if any, making h(x) continuous on [0,5] h(x)={ k cos x 0 ≤ x ≤π 12-x π<x .

   find a value of $k,$ if any, making $h(x)$ continuous on [0,5]
$h(x)=\left\{\begin{array}{ll}k \cos x & 0 \leq x \leq \pi \\ 12-x & \pi<x\end{array}\right.$

Calculus Single Variable

Deborah… 7th Edition

Chapter 1, Problem 49

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Solved by Expert Dwijendra Rao

05/18/2021

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Step 1: We are given a piecewise function $h(x)$ and we need to find the value of $k$ that makes this function continuous on the interval [0,5]. Show more…

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find a value of $k,$ if any, making $h(x)$ continuous on [0,5] $h(x)=\left\{\begin{array}{ll}k \cos x & 0 \leq x \leq \pi \\ 12-x & \pi<x\end{array}\right.$

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Key Concepts

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the domain. They often require special attention at the boundaries where one expression switches to another, as continuity and other properties need to be verified at these transition points.

Continuity at Boundary Points

For a function to be continuous at a point where its defining pieces change, the limit of the function as the variable approaches the boundary point from either side must equal the function's value at that point. This condition leads to equations that must be satisfied for continuity.

Solving for Unknown Parameters in Piecewise Functions

When a piecewise function contains unknown parameters, determining values for these parameters often involves setting the limits from the left and right at the boundary equal. Solving the resulting equation ensures that the function is continuous across the entire domain.

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