# 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 ex 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.