Find the equation of the tangent line to the function h(x) = x/x + 1 when x = 1.

Question

Find the equation of the tangent line to the function h(x) = x/x + 1 when x = 1. Given the function f(x) = x - 1/x + 1, find f'(x) and f"(x). If f(t) = t^2 - 2/t then a) find the average rate of change over the interval 2 to 4. b) next find the instantaneous rate of change when t = 4. Suppose the wholesale price p of calculations and the amount ordered x are related by the equation px + 10x + 5p = 400. If there were ten calculators in stock, currently selling for $20, and the amount ordered is decreasing at a rate of 50%, at what rate is the price changing? Find the equation of the tangent line to the curve f(x) = e^x - e^-x/e^x when x = 0. Find the values of x at which the function h(x) = xe^x has possible extremum and determine if we have a maximum or a minimum at those values. Given the function f(t) = ln(t^2 - e^t), find the derivative when t = 0. A certain material decomposes from 20 grams to 5 grams in 4 years. Find how long for the material to decompose to 10 grams (half-life). If the demand function is given by: q = p/p + 10 - 100, determine if the function is elastic the price is p = 10. Then determine what will happen to revenue if the price is raised. Find the area bounded by the functions: f(x) = 3x + 2 and g(x) = x^2 + 3x + 1. Find the definite integral for: integral_0^1 2x^3 - x^2/3x^4 - 2x^3 + 1 dx

Explanation / Answer

1. h(x) = x/(x+1) when x =1
for the equation of the tangent line we need a point on the tangent and its slope

point on the tangent line is y = h(1) = 1/(1+1)= 1/2
=> the point is (x,h(x)) = (x , 1/2)

now for the slope we'll find the derivative of h(x) as x=1 and that will be the slope
of the tangent line
=> h'(x) = [(x+1) - x]/(x+1)^2 = 1/(x+1)^2
h'(1) = 1/(1+1)^2 =1/4

so we have the point (1,1/2) and the sloope h'(x)= m = 1/4

using the point slope form
(y-y1)= m(x-x1)

=> (y-1/2) = 1/4(x-1)

y = x/4 + 1/4   ----> equation of the tangent line

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