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1. There is the graph of y = 2^x, A(3,8) and B(3.1,8.5741877), the gradient of the chord is 5.74.

Q. Stating the points you use, find the gradient of another chord which will give a closer approximation to the gradient of the tangent to y = 2^x at A.

Is this to do with first principles or something? I'm struggling to understand what this q is asking of me. Thanks!
2. What about trying x = 3.05? That point is closer to A than B.
3. (Original post by Magenta96)
There is the graph of y = 2^x, A(3,8) and B(3.1,8.5741877), the gradient of the chord is 5.74.

Q. Stating the points you use, find the gradient of another chord which will give a closer approximation to the gradient of the tangent to y = 2^x at A.

Is this to do with first principles or something? I'm struggling to understand what this q is asking of me. Thanks!
The gradient at the point x is the limit of the value of (f(x+h)-f(x))/h (note this is the gradient of a chord) as h tends to 0. In this case f(x) is 2^x and the point under consideration is (3,8) which has an x-coordinate of 2.

You have determined the value of the above expression for h=0.1. Due to the shape of the curve, any h smaller than 0.1 and positive will provide a better approximation to the gradient of the curve at (3,8). So go mental, choose any such h, plug it into into the expression and your work here is done!
4. (Original post by Magu1re)
The gradient at the point x is the limit of the value of (f(x+h)-f(x))/h (note this is the gradient of a chord) as h tends to 0. In this case f(x) is 2^x and the point under consideration is (3,8) which has an x-coordinate of 2.

You have determined the value of the above expression for h=0.1. Due to the shape of the curve, any h smaller than 0.1 and positive will provide a better approximation to the gradient of the curve at (3,8). So go mental, choose any such h, plug it into into the expression and your work here is done!
In that expression, so f(x) is 2^x, h is something less than 0.1, but what do I use for f(x+h)? Am I using the x coordinate of 3 and the h < 0.1? I don't get what I'm putting into the expression. I don't know how I'll get rid of the 2^x at the end so I get a number gradient.
5. (Original post by Magenta96)
In that expression, so f(x) is 2^x, h is something less than 0.1, but what do I use for f(x+h)? Am I using the x coordinate of 3 and the h < 0.1? I don't get what I'm putting into the expression. I don't know how I'll get rid of the 2^x at the end so I get a number gradient.
f(x+h) is f, which is the function 2^(whatever is in the brackets after f), evaluated at the input value of x+h, which is a number.

So if h is 0.05 we work out f(3+0.05)=f(3.05)=2^(3.05).

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Updated: April 9, 2013
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