Second Midterm Exam Solutions

Problem One

Compute the derivative with respect to x of

y = 2 sin x cos x

Solution

Problem Two

Use implicit differentiation to compute the derivative of y with respect to x if

Solution Differentiating both sides with respect to x gives

Solving for gives

Problem Three

Use a linear approximation to show that near x = 0

Solution Apply the linearization formula

with and x₀ = 0:

Problem Four

Compute the derivative with respect to x of

Solution The easiest method is logarithmic differentiation.

Differentiating both sides with respect to x gives

Problem Five

One of the lines tangent to the graph of y = e^x passes through the origin. Find the equation of that tangent line.

Solution If the tangent line attaches to the curve at it has slope and equation

The requirement that the tangent line pass through the origin forces

a = 1

The equation of the tangent line is

y = e x