AP Calculus AB: Chapter 6 Practice Test
This set of flashcards focuses on applying implicit differentiation to find derivatives of implicitly defined functions and explores properties of inverse functions. It includes solving for dy/dx in various equations, analyzing tangent lines, and finding or evaluating inverse functions and their derivatives.
Given x^ 2 − y^ 2 = 1, find dy / dx by implicit differentiation.
dy/dx=x/y
Key Terms
Given x^ 2 − y^ 2 = 1, find dy / dx by implicit differentiation.
dy/dx=x/y
Given xy = 5, find dy / dx by implicit differentiation.
−y / x
Given √x+1/√y=1, find dy/dx by implicit differentiation.
√y^3/x
Given cos (x + y) = sin x sin y, find dy / dx by implicit differentiation.
−sin(x+y)+cosxsiny/sin(x+y)+sinxcosy
Evaluate the following as true or false. If dx/dy=1/dydx=0,then the tangent line to the curve y=f(x) is horizontal.
false
Find all points on the curve y + x = x^ 2 + y^2 where the tangent line is horizontal.
(1/2,1+√2/2) and (1/2,1−√2/2)
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| Term | Definition |
|---|---|
Given x^ 2 − y^ 2 = 1, find dy / dx by implicit differentiation. | dy/dx=x/y |
Given xy = 5, find dy / dx by implicit differentiation. | −y / x |
Given √x+1/√y=1, find dy/dx by implicit differentiation. | √y^3/x |
Given cos (x + y) = sin x sin y, find dy / dx by implicit differentiation. | −sin(x+y)+cosxsiny/sin(x+y)+sinxcosy |
Evaluate the following as true or false. If dx/dy=1/dydx=0,then the tangent line to the curve y=f(x) is horizontal. | false |
Find all points on the curve y + x = x^ 2 + y^2 where the tangent line is horizontal. | (1/2,1+√2/2) and (1/2,1−√2/2) |
Find all points on the curve ln (xy) = x^ 2 where the tangent lines are vertical. | None of the tangent lines are vertical. |
Which of the following functions is its own inverse? In other words, for which of the following functions is f −1 (x) = f (x)? | f (x) = 1 / x |
What is the inverse of f(x)=3 3√x+1/5? | f−1(x)=(5x−1/3)^3 |
What is the value of d/dx[f−1(x)] when x=2, given that f(x)=2x−4? | 1/2 |
What is the value of d/dx(f−1(x)) when x=7/3, given that f(x)=x^5+1/3x^3+x and f−1(7/3)=1? | 1/7 |
What is the value of d/dx[f−1(x)] when x=π, given that f(x)=2x+cos^2x and f−1(π)=π/2? | 1/2 |
What does arcsin(√3/2) equal? | π / 3 |
What does arcsin (sin (5π / 4)) equal? | −π / 4 |
Solve the equation arccos (x ^2 − 2x + 2) = 0 for x. | x = 1 |
What is the largest interval containing x = 2π on which sin x is one-to-one? | [3π / 2, 5π / 2] |
Find the derivative d/dx[arctan(x^2+1)] | 2x/1+(x^2+1)^2 |
Find the derivative d/dx[2arcsin(x−1)]. | 2/√−x^2+2x |
Find the derivative of y=l/n(tanhx/2) | csch x |
Find the derivative of f(x)=xcoshx−sinhx | x sinh x |