Learn derivatives calculus and practice equation of tangent line & average rate of change calculus formula. Identifying the given limit definition of the derivative and then evaluating the derivative to get the Limit.
The derivatives of a function \('f'\) at a number \('a'\) is defined as
\(\lim\limits_{h \to 0}\;\left(\dfrac{f(a+h)-f(a)}{h}\right)\) ...(1)
and is denoted by \(f'(a)\) (we are assuming that limit exists).
A \(7\,sin\,a\)
B \(\dfrac{-8}{a^3}\)
C \(\dfrac{200}{a^2}\)
D \(2a\)
\(=\lim\limits_{h\to 0}\,\left(\dfrac{f(a+h)-f(a)}{h}\right)=f'(a)\)
Then, \(f(a+h)=(2+h)^3\)
\(\Rightarrow\,a=2,\;f(x)=x^3\)
\(\therefore\) Limit =\(f'(2)=3x^2\) at \(x=2\)
\(=12\)
A \(a=5,\;f(x)=x^2\)
B \(a=50,\;f(x)=\dfrac{1}{x}\)
C \(a=1,\;f(x)=x^{{1}/{3}}\)
D \(a=3,\;f(x)=sin\,x\)
Let \(y\) be a quantity which is a function of \(x\), \(y=f(x)\). If \(x\) changes from \(x_1\) to \(x_2\) we say.
\(\Delta x =x_2-x_1\) = change in \(x\)
Corresponding change in \(y=f(x_2)-f(x_1)=\Delta y\)
\(=\dfrac{f(x_2)-f(x_1)}{x_2-x_1}=\dfrac{\Delta y}{\Delta x}\)
\(=\lim\limits_{x_2\to x_1}\;\left(\dfrac{f(x_2)-f(x_1)}{x_2-x_1}\right)\)
\(=\lim\limits_{\Delta x\to 0}\;\left(\dfrac{f(x_1+\Delta x)-f(x_1)}{\Delta x}\right)\)
Correlating the above limit with derivative we say, \(f'(a)\) is the instantaneous rate of change of \(y=f(x)\) with respect to \(x\) when, \(x=a\).
A \(\dfrac{7}{8}\)
B \(\dfrac{1}{4}\)
C 19
D –2
e.g. \(\lim\limits_{x\to 2}\;\dfrac{x^{10}-2^{10}}{x-2}=10×2^9\)
because \(f(x)=x^{10},\;a=2\)
\(\therefore\) \(f'(x)\) \(=10x^9\) and \(f'(2)\) \(=10×2^9\)
A \(20×8^{21}\)
B \(50×7^{49}\)
C \(2×7^{8}\)
D \(18×5^{19}\)
The equation of tangent line to \(y=f(x)\) at any point \(P(a,\,f(a))\) on it is a line passing through \(P(a,\,f(a))\) and whose slope is \(f'(a)\).
A \(4x-y-33=0\)
B \(x+2y-1=0\)
C \(x=7\)
D \(y=-8\)
Consider the graph \(y=f(x)\) as shown in figure.
Different tangents at points \(P,\,Q,\;R\) are drawn as shown in figure
We see
At point \(P\)
i.e. at \(x=x_1\), tangent makes acute angle with \(x-\) axis which implies
Slope of tangent \(\to\;f'(x_1)>0\)
at \(x=x_1\)
At point \(Q\)
i.e. at \(x=x_2\), tangent is parallel to \(x-\) axis which implies
Slope of tangent at \(x=x_2\,\to\) \(f'(x_2)=0\)
At point \(R\)
i.e. at \(x=x_3\), tangent makes obtuse angle with \(x-\) axis which implies
Slope of tangent at \(x=x_3\,\to\) \(f'(x_3)<0\)
It also concludes
\(f'(x_1)>f'(x_2)>f'(x_3)\)
A \(f'(2)<f'(4)<f'(6)\)
B \(f'(2)>f'(4)>f'(6)\)
C \(f'(4)<f'(6)<f'(2)\)
D \(f'(6)<f'(2)<f'(4)\)
Let \(y=f(x)\) be a function , if \(x \) changes from \(x _1 \) to \(x _2\), the change in \(x \) (called the increment of \(x \)) is \(\Delta x=x_2 -x_1\)
The corresponding change in \(y\) is \(\Delta y = f(x_2) - f(x_1)\)
The difference quotient = \(\dfrac{\Delta y}{\Delta x} = \dfrac{f(x_2)-f(x_1)}{x_2-x_1}\) is called the average rate of change of \(y\) with respect to \(x\) over the interval \((x_1,x_2)\)
A \(\dfrac{8}{3}\)
B \(\dfrac{7}{3}\)
C \(\dfrac{5}{3}\)
D \(3\)
Average velocity is the average rate of change of position with respect to time during an interval \((t_1,t_2)\) .
Average Velocity = \(\dfrac{\Delta x}{\Delta t} = \dfrac{x(t_2)-x(t_1)}{t_2-t_1}\)
where x=position,t=time