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Questions and Answers
What is the first step in finding dy/dx for the curve defined by $6x^2 + 3xy + 2y^2 + 17y = 96$?
What is the first step in finding dy/dx for the curve defined by $6x^2 + 3xy + 2y^2 + 17y = 96$?
- Isolate y on one side of the equation
- Differentiate both sides with respect to x (correct)
- Complete the square for the equation
- Evaluate the equation at specific points
At the point (4,0), what is the slope of the tangent line to the curve $6x^2 + 3xy + 2y^2 + 17y = 96$?
At the point (4,0), what is the slope of the tangent line to the curve $6x^2 + 3xy + 2y^2 + 17y = 96$?
- 0
- -1
- 2 (correct)
- 3
What is the purpose of using the tangent line to approximate the value of k at the point (3.8, k)?
What is the purpose of using the tangent line to approximate the value of k at the point (3.8, k)?
- To create an estimate of k based on nearby values (correct)
- To determine the slope of the curve at that point
- To simplify the computation of k
- To find the exact value of k
Which of the following statements about the function g(x) = x·f(x) is true regarding the closed interval [-1, 5]?
Which of the following statements about the function g(x) = x·f(x) is true regarding the closed interval [-1, 5]?
According to the Intermediate Value Theorem, what can be concluded about g(x) in the interval (1, 3)?
According to the Intermediate Value Theorem, what can be concluded about g(x) in the interval (1, 3)?
What is the expression for dy/dx at the point (4,0) for the curve defined by $6x^2 + 3xy + 2y^2 + 17y = 96$?
What is the expression for dy/dx at the point (4,0) for the curve defined by $6x^2 + 3xy + 2y^2 + 17y = 96$?
What is the equation of the tangent line to the curve at the point (4,0)?
What is the equation of the tangent line to the curve at the point (4,0)?
Using the tangent line approximation, what is the estimated value of k for the point (3.8,k)?
Using the tangent line approximation, what is the estimated value of k for the point (3.8,k)?
What equation can be solved to find the actual value of k such that the point (3.8,k) is on the curve?
What equation can be solved to find the actual value of k such that the point (3.8,k) is on the curve?
What can be deduced about g(x) concerning the value -2 based on the Intermediate Value Theorem within the interval (1, 3)?
What can be deduced about g(x) concerning the value -2 based on the Intermediate Value Theorem within the interval (1, 3)?
Flashcards
Tangent Line Equation
Tangent Line Equation
The equation of the line that touches a curve at a single point, representing the instantaneous rate of change at that point.
Implicit Differentiation
Implicit Differentiation
A technique for finding the derivative of a function when the function isn't explicitly defined as y = f(x).
Average Rate of Change
Average Rate of Change
The overall change in a function over an interval divided by the length of the interval.
Intermediate Value Theorem
Intermediate Value Theorem
If a function is continuous on a closed interval and takes on two values, it must also take on every value in between.
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Inverse Function Derivative
Inverse Function Derivative
The derivative of the inverse function at a point is the reciprocal of the derivative of the original function at the corresponding point on the inverse function's graph.
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Tangent Line Equation
Tangent Line Equation
The equation of a line that touches a curve at a single point.
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Implicit Differentiation
Implicit Differentiation
Finding a derivative when the function isn't written as y=f(x).
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Average Rate of Change of g(x)
Average Rate of Change of g(x)
Overall change in g(x) over an interval, divided by the interval length.
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Intermediate Value Theorem
Intermediate Value Theorem
If a continuous function takes on two values in an interval, it must also take on all values between.
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Inverse Function Derivative, h’(-2)
Inverse Function Derivative, h’(-2)
Derivative of the inverse function at a point, related to the derivative of the original function.
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Calculus AB - Section II, Part B
- Calculator not allowed for this part
- Show all work and justify answers
Problem 1
- Curve equation: 6x² + 3xy + 2y² + 17y = 96
- Part (a): Find the derivative dy/dx
- Part (b): Find the equation of the tangent line at (4,0)
- Part (c): Use the tangent line to approximate k for point (3.8,k) on the curve.
- Part (d): Find the actual value of k so that (3.8,k) is on the curve (equation)
Problem 2
- Graph of f(x) given for the closed interval -1 ≤ x ≤ 5
- Horizontal tangents: at x = 1, x = 3
- f’(2) = -2
- Part (a): g(x) = x * f(x). Find the equation of the tangent line at x = 2 for g(x)
- Part (b): Find the average rate of change of g(x) for -1 ≤ x ≤ 5
- Part (c): Show that g(x) = -2 on the interval 1 < x ≤ 3 using the Intermediate Value Theorem
- Part (d): h(x) is the inverse of g(x). Find h’(-2)
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