Learn slope of tangent and secant line, average and instantaneous velocity formula calculus. Practice to find the slope of the curve at the given point.

Consider a curve \(C\), and two points \(A(a,\,f(a))\) and \(B(x,\,f(x))\) on it, the curve being \(y=f(x)\)

slop of secant line AB \(\dfrac{f(x)-f(a)}{x-a}\)

slope of secant line AB = \(\dfrac{f(b)-(f(a)}{b-a}\)

- Slope of line joining \((x_1,\,y_1)\) and \((x_2,\,y_2)\) is

\(m=\dfrac{y_2–y_1}{x_2–x_1}\)

slope of tangent at A= m_{t }= \(\lim\limits_{x\to a} \dfrac{f(x)-f(a)}{x-a}\)

As B approaches A, this secant AB will become tangent at A.

We are assuming that the above limit exists.

The equation of a straight line which passes through a fixed point \((x_1,\,y_1)\) and whose slope is \(m\) is given by,

\(y-y_1=m(x-x_1)\)

\(m=tan\theta=\dfrac{y-y_1}{x-x_1}\)

- A line is completely determined if its slope and a fixed point is given on it.

A \(x+y+8=0\)

B \(2x-y+11=0\)

C \(2x=3\)

D \(y=7\)

The slope of tangent to a curve \(y=f(x)\) at the point \(A(a,\,f(a))\) is

\(m_t=\lim\limits_{x\to a}\,\left(\dfrac{f(x)-f(a)}{x-a}\right)\)

A 4

B –4

C 5

D \(\dfrac{1}{5}\)

\(m_t=\lim\limits_{x\to a}\,\left(\dfrac{f(x)-f(a)}{x-a}\right)\)

Let, \(x-a=h\)

\(\therefore\) as \(x\to a\)

\(\Rightarrow\,h\to 0\)

\(\Rightarrow\,m_t=\lim\limits_{h\to 0}\,\dfrac{f(a+h)-f(a)}{h}\)

- This limit will be the slope of tangent only if it exists.

A \(y=-3\)

B \(x=8\)

C \(x+2y-1=0\)

D \(2x-5y+3=0\)

A particle moves in a straight line with displacement function, \(s=f(t)\) at the time \('t'\) then

- \(\text{Average velocity}=\dfrac{f(t_2)-f(t_1)}{t_2-t_1}=\dfrac{\text{displacement}}{\text{time}}\)

\(=\dfrac{f(a+h)-f(a)}{h}\to (t_1=a\,\text{ and}\,\,t_2=a+h)\)

\(\Rightarrow\,\text{Average velocity}=\dfrac{f(a+h)-f(a)}{h}\)

A \(78\,m/s\)

B \(-2m/s\)

C \(0.5m/s\)

D \(16m/s\)

The instantaneous velocity \(v(a)\) at the time \(t=a\) is defined as limit of average velocity.

\(v(a)=\lim\limits_{h\to 0}\;\left(\dfrac{f(a+h)-f(a)}{h}\right)\)

where, \(f(t)\) = displacement at time \('t'\).

- Instantaneous velocity will be simply referred to as velocity at \(t=a\).

A \(-7\,m/s\)

B \(9\,m/s\)

C \(12\,m/s\)

D \(18\,m/s\)