Implicit Differentiation (College Board AP® Calculus AB): Exam Questions

Consider the curve given by the equation for .

Show that .

Write an equation for the line tangent to the curve at the point .

A celestial body changes its temperature, degrees Fahrenheit, over time, minutes, according to the relationship:

Find the rate at which the temperature of the body changes, .

Is the celestial body cooling faster when it is or when it is Explain your reasoning.

Consider the curve given by the equation

Find an expression for in terms of and

Find the slope of the tangents to the curve at the two points where the curve intersects the -axis.

Show that the derivative of is .

Show the work that leads to your answer.

Consider the curve given by the equation

Find an expression for in terms of and

The curve defined by passes through the point

Write an equation for the line normal to the curve at the point express your final answer in slope-intercept form.

Consider the curve given by the equation .

Show that .

A particle is moving along the curve. At the instant when the particle is at the point , its horizontal position is increasing at a rate of units per second.

What is the value of , the rate of change of the particle's vertical position, at that instant?

Use implicit differentiation to show that the derivative of is .

Show all steps of your working.

Show that if then

A treasure chest is being raised from the bottom of the ocean using a pulley system. The depth of the chest below the ocean surface, in meters, and the length of rope unwound from the pulley, in meters, are related by the equation:

where is the time elapsed, in minutes, since the pulley started operating.

Show that and explain what this represents in the context of the problem.

At minutes, the chest is at a depth of meters and the length of the rope is changing at a rate of meters per minute.

Find the rate at which the depth of the treasure chest is changing at minutes. Explain whether this means the treasure chest is rising or falling at this time.

At minutes the length of the rope is now meter and the rate at which the length of rope is changing has not changed. Determine whether the depth of the chest is changing faster or slower than at minutes.

Consider the curve given by the equation

Show that .

Verify that the point with coordinates lies on the curve with equation

Then, find the equation of the tangent to the curve at this point. Express your final answer in slope-intercept form.

The tangent line to the curve with equation at the point with coordinates intercepts the -axis at the point and the -axis at the point Find the area of the triangle formed by the origin, and the points and