Learn example of mean value theorem problems for derivatives and practice Newton Raphson Method in calculus, Use Newton's Method to find approximate value.
Let \(f\) be a function defined in [a, b] such that
then, there exists at least one number \(c\in(a,b)\) such that
\(f'(c)=\dfrac {f(b)-f(a)}{b-a}\) or \((b-a)\;f'(c)=f(b)-f(a)\)
A \(f(x) = 4x^3+7x^2-x+1\) in [–2, 5 ]
B \(f(x) = sinx+2\) in [5, 7 ]
C \(f(x) = tanx\) in \([0, \pi ]\)
D \(f(x)=\dfrac {1}{x}\) in [5, 9 ]
Step 1 : Find \(f'(x)\) i.e. derivative of \(f(x)\).
Step 2 : Put\( f ' (x) = \dfrac {f(b) - f(a)}{b-a}\)
Step 3 : Find roots of the above equation, these will be the values of 'c'
A c = –1
B c = 3
C c = 4/7
D c = 5/2
\(y-f(x_1)=f'(x_1)(x-x_1)\;...(i)\)
this will intersect \(x\) axis at \(x_2\) (Put \(y=0\))
\(-f(x_1)=f'(x_1)(x_2-x_1)\)
\(\Rightarrow\;x_2=x_1-\dfrac {f(x_1)}{f'(x_1)} \Rightarrow\;x_3=x_2-\dfrac {f(x_2)}{f'(x_2)}\)
\(\Rightarrow\;x_{n+1}=x_n-\dfrac {f(x_n)}{f'(x_n)} \)
We can find better approximate by increasing values of \(n\).
A \(x_3=1.75\)
B \(x_3=8.34\)
C \(x_3=2.76\)
D \(x_3=1.09\)
A \(x_4=1.893\)
B \(x_4=.962\)
C \(x_4=5.372\)
D \(x_4=1.125\)
A c =\(2\pm\dfrac {2}{\sqrt 3}\)
B c = \(4\pm\dfrac {2}{\sqrt 3}\)
C c = \(1\pm\sqrt 3\)
D c = \(\pm5\)
By LMVT,
\(f'(c)= \dfrac {f(b)-f(a)}{b-a}\)
for some \(c\in(a,b)\)
A \(x^3-12x+2=y\) has a horizontal tangent at x = 2
B \(x^3-12x+2=y\) has a vertical tangent at x = 2
C \(x^3-12x+2=y\) has a discontinuous at x = 2
D \(x^3-12x+2=y\) has a non differentiable at x = 2
A \(m= 9, \,n=19\)
B \(m= 90, \,n=9\)
C \(m= 19, \,n=9\)
D \(m= 9, \,n=9\)