Lesson Tangent lines

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A function y = f(x) and an x-value are given.
(a) Find a formula for the slope of the tangent line to the graph of f at a general point x=x%5B0%5D
(b) Use the formula obtained in part (a) to find the slope of the tangent line for the given value of x%5B0%5D
16.
a. f%28x%29=x%5E2%2B3x%2B2%3B+x%5B0%5D=2
lim%28x-%3Ex%5B0%5D%2C%28f%28x%29-f%28x%5B0%5D%29%29%2F%28x-x%5B0%5D%29%29 use this formula to evaluate the limit to find the slope of the tangent line.
distribute the negative
lim%28x-%3Ex%5B0%5D%2C%28x%5E2%2B3x%2B2-x%5B0%5D%5E2-3x%5B0%5D-2%29%2F%28x-x%5B0%5D%29%29 the 2s cancel
lim%28x-%3Ex%5B0%5D%2C%28x%5E2%2B3x-x%5B0%5D%5E2-3x%5B0%5D%29%2F%28x-x%5B0%5D%29%29 rearrange the terms
lim%28x-%3Ex%5B0%5D%2C%28x%5E2-x%5B0%5D%5E2%2B3x-3x%5B0%5D%29%2F%28x-x%5B0%5D%29%29 factor using difference of squares
cancel all x-x%5B0%5D factors
lim%28x-%3Ex%5B0%5D%2C%28x%2Bx%5B0%5D%2B3%29%29 now substitute x%5B0%5D for x
x%5B0%5D%2Bx%5B0%5D%2B3
2x%5B0%5D%2B3 this is the general formula for the slope of the tangent line at any x%5B0%5D
b. at x%5B0%5D=2
mtan=2%282%29%2B3=7 the slope of the tangent line at x=2 is 7.
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