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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