Advertisements
Advertisements
प्रश्न
In the following, determine the value of constant involved in the definition so that the given function is continuou: \[f\left( x \right) = \begin{cases}5 , & \text{ if } & x \leq 2 \\ ax + b, & \text{ if } & 2 < x < 10 \\ 21 , & \text{ if } & x \geq 10\end{cases}\]
Advertisements
उत्तर
Given:
\[ \Rightarrow \lim_{h \to 0} \left( 5 \right) = \lim_{h \to 0} \left[ a\left( 2 + h \right) + b \right] \text{ and } \lim_{h \to 0} \left[ a\left( 10 - h \right) + b \right] = \lim_{h \to 0} \left( 21 \right)\]
\[ \Rightarrow 5 = 2a + b . . . \left( 1 \right) \text{ and } 10a + b = 21 . . . \left( 2 \right)\]
\[\text{ On solving eqs } . \left( 1 \right) \text{ and } \left( 2 \right), \text{ we get } \]
\[a = 2 \text{ and } b = 1\]
APPEARS IN
संबंधित प्रश्न
Examine the continuity of the following function :
`{:(,,f(x)= x^2 -x+9,"for",x≤3),(,,=4x+3,"for",x>3):}}"at "x=3`
If 'f' is continuous at x = 0, then find f(0).
`f(x)=(15^x-3^x-5^x+1)/(xtanx) , x!=0`
If f(x)= `{((sin(a+1)x+2sinx)/x,x<0),(2,x=0),((sqrt(1+bx)-1)/x,x>0):}`
is continuous at x = 0, then find the values of a and b.
Examine the following function for continuity:
f(x) = x – 5
Discuss the continuity of the function f, where f is defined by:
f(x) = `{(-2", if" x <= -1),(2x", if" -1 < x <= 1),(2", if" x > 1):}`
If \[f\left( x \right) = \begin{cases}e^{1/x} , if & x \neq 0 \\ 1 , if & x = 0\end{cases}\] find whether f is continuous at x = 0.
Discuss the continuity of the following function at the indicated point:
`f(x) = {{:(|x| cos (1/x)",", x ≠ 0),(0",", x = 0):} at x = 0`
Discuss the continuity of the following functions at the indicated point(s):
Discuss the continuity of the following functions at the indicated point(s):
Show that
\[f\left( x \right) = \begin{cases}1 + x^2 , if & 0 \leq x \leq 1 \\ 2 - x , if & x > 1\end{cases}\]
Determine the value of the constant k so that the function
\[f\left( x \right) = \left\{ \begin{array}{l}\frac{x^2 - 3x + 2}{x - 1}, if & x \neq 1 \\ k , if & x = 1\end{array}\text{is continuous at x} = 1 \right.\]
Find the value of k for which \[f\left( x \right) = \begin{cases}\frac{1 - \cos 4x}{8 x^2}, \text{ when} & x \neq 0 \\ k ,\text{ when } & x = 0\end{cases}\] is continuous at x = 0;
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point;
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}k x^2 , & x \geq 1 \\ 4 , & x < 1\end{cases}\]at x = 1
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \binom{\frac{x^3 + x^2 - 16x + 20}{\left( x - 2 \right)^2}, x \neq 2}{k, x = 2}\]
Define continuity of a function at a point.
The function \[f\left( x \right) = \begin{cases}\frac{e^{1/x} - 1}{e^{1/x} + 1}, & x \neq 0 \\ 0 , & x = 0\end{cases}\]
If \[f\left( x \right) = \frac{1}{1 - x}\] , then the set of points discontinuity of the function f (f(f(x))) is
Show that f(x) = |x − 2| is continuous but not differentiable at x = 2.
Show that \[f\left( x \right) =\]`{(12x, -,13, if , x≤3),(2x^2, +,5, if x,>3):}` is differentiable at x = 3. Also, find f'(3).
Show that the function
\[f\left( x \right) = \begin{cases}\left| 2x - 3 \right| \left[ x \right], & x \geq 1 \\ \sin \left( \frac{\pi x}{2} \right), & x < 1\end{cases}\] is continuous but not differentiable at x = 1.
Discuss the continuity and differentiability of f (x) = |log |x||.
If \[f\left( x \right) = \begin{cases}\frac{1}{1 + e^{1/x}} & , x \neq 0 \\ 0 & , x = 0\end{cases}\] then f (x) is
Let \[f\left( x \right) = \begin{cases}1 , & x \leq - 1 \\ \left| x \right|, & - 1 < x < 1 \\ 0 , & x \geq 1\end{cases}\] Then, f is
Find whether the following function is differentiable at x = 1 and x = 2 or not : \[f\left( x \right) = \begin{cases}x, & & x < 1 \\ 2 - x, & & 1 \leq x \leq 2 \\ - 2 + 3x - x^2 , & & x > 2\end{cases}\] .
Evaluate :`int Sinx/(sqrt(cos^2 x-2 cos x-3)) dx`
Discuss the continuity of f at x = 1
Where f(X) = `[ 3 - sqrt ( 2x + 7 ) / ( x - 1 )]` For x ≠ 1
= `-1/3` For x = 1
If f is continuous at x = 0, then find f (0).
Where f(x) = `(3^"sin x" - 1)^2/("x" . "log" ("x" + 1)) , "x" ≠ 0`
Find the points of discontinuity , if any for the function : f(x) = `(x^2 - 9)/(sinx - 9)`
Examine the continuity off at x = 1, if
f (x) = 5x - 3 , for 0 ≤ x ≤ 1
= x2 + 1 , for 1 ≤ x ≤ 2
If the function f is continuous at x = I, then find f(1), where f(x) = `(x^2 - 3x + 2)/(x - 1),` for x ≠ 1
Find the value of the constant k so that the function f defined below is continuous at x = 0, where f(x) = `{{:((1 - cos4x)/(8x^2)",", x ≠ 0),("k"",", x = 0):}`
If f(x) = `{{:((x^3 + x^2 - 16x + 20)/(x - 2)^2",", x ≠ 2),("k"",", x = 2):}` is continuous at x = 2, find the value of k.
Examine the differentiability of the function f defined by
f(x) = `{{:(2x + 3",", "if" -3 ≤ x < - 2),(x + 1",", "if" -2 ≤ x < 0),(x + 2",", "if" 0 ≤ x ≤ 1):}`
The number of points at which the function f(x) = `1/(x - [x])` is not continuous is ______.
f(x) = `{{:((2x^2 - 3x - 2)/(x - 2)",", "if" x ≠ 2),(5",", "if" x = 2):}` at x = 2
f(x) = `{{:(("e"^(1/x))/(1 + "e"^(1/x))",", "if" x ≠ 0),(0",", "if" x = 0):}` at x = 0
