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anonfawkes anonfawkes
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Posts: 14
Rep: 1 0
11 years ago
So, I've tried for over an hour to figure some way for these 2 problems to make sense to me and I haven't gotten anywhere except halfway through a jar of Nutella crying over how much I suck at this (and I'm minoring in math). Can someone please help? Maybe work out the 1st one completely & show all steps then I'll try the other one on my own based on what your example shows? I don't know. I just can't focus any longer.

1. Let f(x) = sin(x).
(a) Sketch the graph of f over the interval x ∈ [−π/2,π/2]. Use a 1-1 scale (1 unit in the
horizontal is as long as 1 unit in the vertical).
(b) Estimate the slope of the line tangent to the graph at x0 = 0 as follows. Make a table whose first row contains a set of values of h (of your choice) approaching zero, and whose second row contains the slopes of the secant lines through (x0,f(x0)) and (x0 + h,f(x0 + h)). Then state the estimated limiting tangent slope.
(c) Find the equation for the tangent line to the graph of f at x = x0. Add a sketch of the tangent line and one sample secant line for one value of h in your figure in (a).


2. Let f(x) = x3/2.
(a) Sketch the graph of f over the interval x ∈ [0, 2]. Use a 1-1 scale.
(b) Estimate the slope of the line tangent to the graph at c = 1 as follows. Make a table whose first row contains a set of special values of x (of your choice) approaching c, and whose second row contains the slopes of the secant lines through (c,f(c)) and (x,f(x)). Then state the estimated limiting tangent slope.
(c) Find the equation for the tangent line to the graph of f at x = c. Add a sketch of the tangent line and one sample secant line for one special values of x in your figure in (a).
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11 years ago
Don't hesitate to present your progress when making a question. Here are some points that may help:

I guess 1 a) shouldn't be hard for you. Here is an image (you have to restrict the graph to x= (-π/2,π/2) )
b)
Firstly let's find the slope of line that goes through (x0,f(x0)) and (x0 + h,f(x0 + h)). x0=0 so f(x0)=0.
So we need to find the slope of the line that goes through (0,0) and (h,f(h)). The slope is: [(f(h)-0)/(h-0)]=[f(h)/h].
Now: you choose the values of h (values that approaches 0 like π/6 and -π/6, or more closer to zero) and you find the slopes of these lines by replacing h with these numbers. (for example f(π/6)/(π/6)=...) By choosing numbers close enough to zero, you can estimate the slope of the tangent. For example, if you find the slope for h=0,01 the valuse will probably be close with the slope of the tangent. Why's that happening? Because the slope of the tangent is the limit as x approaches 0.

c) To find the exact equation of the tangent, you will have to calculate the slope of the tangent accurately. The slope of the tangent at (0,0) is no other than: f ' (0)= cos(0)= 1. So the equation of the tangent is y=1*x + 0 <=> y=x.
Then you just sketch this line, and two more lines (for two h values). That should look like (see attachment. The black line represent the tangent.)

The second problem is similar to the first, but it uses a different function (f(x)=x3/2 )
I suggest you to try to solve that one and if you can, you may post again showing your progress or ask anything you want.
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