first day of calculus. (1 Viewer)

[elliottjohnson]

hey im in year 11 and we just started the wonderful world of calculus yesterday. can you please help me out this this quick question i cant seem to get the right answer which is -13.

f(x) = 3x^2 - 7x - 4 at the point (-1, 6)

Differentiate from first principles to find the gradient of the tangent to the curve.

ive heard you dont use first principle much but it can crop up in your HSC.

thanks for any help if you can show workings.

 

[annabackwards]

f(x) = 3x^2 - 7x - 4
f(x+h) = 3(x+h)^2 - 7(x+h) - 4

Gradient
= lim [ f(x+h) - f(x) ] / h
(h --> 0)

= lim [3(x+h)^2 - 7(x+h) - 4 - (3x^2 - 7x - 4 ) ] / h
(h --> 0)

= lim [3(x^2+ 2xh + h^2) - 7x -7h - 4 - 3x^2 + 7x + 4 ) ] / h
(h --> 0)

= lim [6xh + 3h^2 -7h] / h
(h --> 0)

= lim h[6x + 3h -7 ] / h
(h --> 0)

= lim [6x + 3h -7 ]
(h --> 0)

= 6x - 7

At (-1, 6), gradient = 6(-1) - 7 = -13

 

[elliottjohnson]

thanks! thats great.

 

[annabackwards]

No worries.

Remember that you can always check your answer using normal differentiation, which you'll learn soon

 

[Timothy.Siu]

lol
i like how it said "the wonderful world of calculus"

 

[Lukybear]

TO OP: Youll learn soon enough.

 

[TheStallion]

Yep, learning the first, long method is always a bitch, then when you learn normal differentiation, it all becomes so much easier.

 

[jet]

Yeah, it pops up in the HSC, but they always give you the formula.