find the slope of the graph?

Question

f(x)=x^2-x at the point (a,f(a))

Answer 1

I think your teacher wants you to use the definition of the derivative to do this problem.

Let's use f'(a)=\lim_{x \to a) [f(x)-f(a)]/(x-a).

After factoring, be sure to divide carefully in the following: (a+b)/c=a/c+b/c.

Then,

f'(a)=\lim_{x \to a) [f(x)-f(a)]/(x-a)
=\lim_{x \to a) (x^2-x-a^2+a)/(x-a)
=\lim_{x \to a) [(x-a)(x+a)-(x-a)]/(x-a)
=\lim_{x \to a) [(x+a)-1]
=2a-1

You will get the same result if you use f'(a)=\lim_{h \to 0} [f(a+h)-f(a)]/h and should try it to make sure you can get the calculation right.

Your teacher will expect you to be able to use the definition of the derivative to find the derivative of a low degree polynomial in a testing situation, so you should practice a lot and make sure that you are able to do so.

Answer 2

think of this as the summation of the curves
x^2 (a parabola) and a straight line -x

The slope of x^2 is 2x (this is logical since this graph looks like a cup with slopes increasing fast the more it goes away from 0,0.

at point a, the slope is 2a

The slope of y=-x is -1 (at any point)

Sum these two slopes and you get 2a - 1 without calculus.

Answer 3

Well,
f(x) = x^2 - x
so f'(x) (differential of x) = 2x - 1
If x = a then gradient at a is hence 2a - 1

Answer 4

Take the deriv:

f'=2x-1

Slope = 2a-1 (Replace x with a)

Answer 5

differentiate the function f(x) you get
f'(x) = 2x-1
at point a, you get
f'(a) = 2a-1

then the slope is 2a-1.

Answer 6

well, we have 3 correct answers above

Answer 7

interesting