Solved Problems for Derivatives: Applications

Here we will show some typical applications that use derivatives.
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Problem: Find the tangent line and the normal line at a = 1 to the graph of the function
Problem: Where is the tangent line to the graph of

f (x) = x³ − 3x² − 9x + 11

horizontal?
Problem: Find the global extrema of

f (x) = x² − 8|x − 1| + 8

on the interval I = [−1,6].
Problem: Find the global extrema of

on the interval I = [1,∞).
Problem: A beverage is to be sold in cans shaped as cylinder. The volume of this cylinder must be 350 cm³. What dimensions will minimize its surface (and therefore the cost of material)?
Problem: On a straight shoreline there is a tree and exactly opposite it, 100 m away in the sea, stands a lighthouse. A strong and thin spotlight on its top revolves at the rate of one revolution per 4 seconds, its light creating a running light spot on the shore. You stand on the shore 100 m from the tree. How fast does this spot move when it goes past you?
Problem: Find the Taylor polynomial of degree 3 and of degree n with center a = 0 for the function

f (x) = ln(1+2x).
Problem: Use tangent line to find an approximating formula for

for x close to 1.
Problem: Use second degree Taylor polynomial to estimate cos(0.5).
Estimate the error of this approximation.
Problem: We replace e^x by its Taylor polynomial of degree 60 with center a = 0. Estimate the error of approximation that we make if we use this for x between 0 and 20. Estimate the relative error of approximation there.