Given:
The equation of the curve is:
[tex]y=2x^3-5x+1[/tex]
To find:
The gradient (slope) of the given curve at point (2,7).
Solution:
We have,
[tex]y=2x^3-5x+1[/tex]
Differentiate the given equation with respect to x.
[tex]y=2(3x^2)-5(1)+(0)[/tex]
[tex]y'=6x^2-5[/tex]
Now we need to find the value of this derivative at (2,7).
[tex]y'_{(2,7)}=6(2)^2-5[/tex]
[tex]y'_{(2,7)}=6(4)-5[/tex]
[tex]y'_{(2,7)}=24-5[/tex]
[tex]y'_{(2,7)}=19[/tex]
Therefore, the gradient (slope) of the given curve at point (2,7) is 19.
