Homework SourseLet fx 12x1 Find f using limit definition Find slope of tan
Let f(x) = 1/2x+1
Find f\' using limit definition
Find slope of tangent line to y=f(x) at x=1
Find the equation of the tangent line to y=f(x) through (1,f(1))
Sketch graph
Please show and explain!!
Find f\' using limit definition
Find slope of tangent line to y=f(x) at x=1
Find the equation of the tangent line to y=f(x) through (1,f(1))
Sketch graph
Please show and explain!!
Solution
RHL
f\' = lim h->0 (f(x+h)-f(x))/h
f\' = lim h ->0 (1/2(x+h) +1 - 1/2x -1)/h
f\' = lim h->0 (1/2(x+h) -1/2x)/h
f\' = lim h->0 (x-x-h)/(2(x+h)(x)(h))
f\' = lim h->0 -h/(2(x+h)(x)(h))
f\' = lim h->0 -1/2(x+h)x
f\' = lim h-> 0 -1/2x^2
LHL
f\' = lim h->0 (f(x)-f(x-h))/h
f\' = lim h ->0 (1/2(x) +1 - 1/2(x-h) -1)/h
f\' = lim h->0 (1/2(x) -1/2(x-h))/h
f\' = lim h->0 (x-h-x)/(2(x-h)(x)(h))
f\' = lim h->0 -h/(2(x-h)(x)(h))
f\' = lim h->0 -1/2(x-h)x
f\' = lim h-> 0 -1/2x^2
So f\' = -1/2x^2
Hence slope at x = 1 is -1/2*1^2 = -1/2
