
Step-by-step solution-:
Simplify the given expression for x=2:
y=x^2+2x+5
Simplify the given expression for x=2+h:
y=(2+h)^2+2(2+h)+5
Let be m a slope of line PQ:
The slope formula is:
m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}
Substitute the coordinate of points A(2,13) and B(2+h, h^2+6h+13) into the slope formula:
m=\frac{h^2+6h+13-13}{2+h-2}
After simplifying we get,
m=\frac{h^2+6h}{h}
m=\frac{\cancel{h}(h+6)}{\cancel{h}}
After simplifying we get,
m=h+6
We know that the first derivative of any curve at the point (x,y) is equal to the slope of the tangent:
It means,
m=\frac{dy}{dx}
Substitute the y=x^2+2x+5 into the above:
m=\frac{d(x^2+2x+5)}{dx}=2x+2
Substitute the y=2 into the above:
m=2\times2+2
After simplifying we get,
m=6
The required value of m is 6.
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