AP Calculus AB - Implicit Differentiation and Related Rates

Flashcards covering implicit differentiation, related rates, and tangent line approximations, based on the provided lecture notes, for AP Calculus AB review.

What rule is essential for implicit differentiation?

The chain rule.

What are the applications of derivatives discussed?

Related rates problems, velocity, and acceleration.

If , then what is the value of at the point ?

Undefined

If cos (xy) = y-1, then what is the value of dy/dx when and y = 1?

0

If ex − y = xy3 + e2 − 18, what is the value of dy/dx at the point (2, 2) ?

Answer C This option is correct. During the implicit differentiation, both the product rule and the chain rule are needed.

If 3x2 + 2xy + y2 = 1, then = ?

(A) (B) (C) (D) (E)

If , then when x=-1, is ?

(A) (B) (C) -1 (D) -2 (E) nonexistent

In related rates, what does the chain rule allow us to do?

Relate the rates of change of different variables.

If x2+xy=10, then when what is dy/dx?

(A) (B) -2 (C) (D) (E)

If , then in terms of x and y, what is dy/dx?

(A) (B) (C) (D) (E)

If sin x = ey, 0 < x < π , what is dy/dx in terms of x?

cot x

If y=ln(x2+y2) , then the value of at the point (1, 0) is

2

If x2+y2=25, what is the value of at the point (4,3) ?

-4/3

The points and are on graph of a function that satisfies the differential equation . Which of the following must be true?

is a local minimum of f.

What is the slope of the line tangent to the curve at the point ?

Answer B Correct. The slope of the tangent line is the value of at the point. During the implicit differentiation to find , both the power rule and the chain rule are needed. The point is on the curve, since and satisfy the equation . At this point, .

If , then the value of at the point is

Answer C Correct. The point is on the curve because , satisfies the implicit equation of the curve. The chain rule is the basis for implicit differentiation.

If , then

Answer D Correct. The chain rule is the basis for implicit differentiation as well as the differentiation of a composite function.

If , then

Answer B Correct. The chain rule is the basis for implicit differentiation.

If , then the value of at the point is

Answer A Correct. The point is on the curve because , satisfies the implicit equation of the curve. The chain rule is the basis for implicit differentiation.

If , then

Answer D Correct. The chain rule is the basis for implicit differentiation as well as the differentiation of a composite function.

If , then

Answer C Correct. The chain rule is the basis for implicit differentiation.

If , then ⅆ ⅆ

Answer A Correct. Both the product rule and the chain rule are used during the implicit differentiation to find the value of ⅆ ⅆ . ⅆ ⅆ ⅆ ⅆ ⅆ ⅆ ⅆ ⅆ ⅆ ⅆ ⅆ ⅆ

What is the slope of the line tangent to the curve at the point ?

Answer C Correct. Both the product rule and the chain rule are used during the implicit differentiation to find the value of at the point .

If what is the value of at the point

Undefined

Let y = f (x) be a twice-differentiable function such that f (1) = 2 and . What is the value of at x = 1 ?

Answer C This option is correct. The answer is obtained by using the chain rule when doing implicit differentiation.The second derivative is found in terms of y and , which can then be written completely in terms of y.The condition f (1) = 2 means that y = 2 when x = 1.

The slope of the tangent to the curve at (2, 1) is

-1

Functions , , and are differentiable with respect to time and are related by the equation . If x is decreasing at a constant rate of 1 unit per minute and y is increasing at a constant rate of 4 units per minute, at what rate is w changing with respect to time when and ?

Answer A Correct. One way to verify that a function is a solution to a differential equation is to check that the function and its derivatives satisfy the differential equation. The differential equation in this option involves and . The correct derivative must be computed and the algebra correctly done to verify that the differential equation is satisfied.

At time , a cube has volume and edges of length . If the volume of the cube decreases at a rate proportional to its surface area, which of the following differential equations could describe the rate at which the volume of the cube decreases?

(E)

Description of removable discontinuity

A removable discontinuity occurs at if exists, but does not exist or is not equal to the value of the limit.

How a tangent line approximates function's value?

tangent line value at will be an underestimate if the graph of is concave up on the interval .

What does the value of when would be an approximation to?

Would be an approximation to , as follows.

Tangent line to the graph of at

An equation of the line tangent to the graph of at is .