Quiz Flashcards
You better do well or I'm going to beat you up
Find the limit of (2x^3 - 5x^2 + 3x + 1) / (x^2 - 4x + 3) as x approaches 1.
The limit is 1
Determine the equation of the tangent line to the curve y = x^3 + 2x^2 - 5x + 1 at the point where x = 2.
The equation of the tangent line is y = 13x - 17.
Find the derivative of f(x) = (4x^2 - 3x + 2) / (5x - 1).
The derivative is f’(x) = (20x + 3)/(5x - 1)^2.
Evaluate the integral of e^(-x^2) with respect to x.
The integral of e^(-x^2) does not have a standard form in terms of elementary functions but is closely related to the error function, denoted as erf(x).
Calculate the second derivative of g(x) = sin(x) / x.
The second derivative is g’‘(x) = (2 - 2sin(x)/x - cos(x)/x^2).
Determine the absolute maximum and minimum values of the function f(x) = x^3 - 6x^2 + 9x + 2 on the interval [0, 4].
The absolute maximum is f(0) = 2, and the absolute minimum is f(3) = -4.
Find the derivative of y = (3x^2 + 2x)(4x - 1)^3.
The derivative is y’ = 3(4x - 1)^3 + (3x^2 + 2x)(12x - 3).
Evaluate the integral of (2x + 1) / (x^2 + 3x + 2) with respect to x.
The integral evaluates to ln|2x + 1| - ln|x + 2| + C.
Find the equation of the line tangent to the curve y = ln(4x) at the point where x = 1.
The equation of the tangent line is y = x - 1.
Determine the average value of the function f(x) = x^2 on the interval [1, 3].
The average value is 7/3.
Find the derivative of y = (x^2 - 1)(x^3 + 2x + 1).
The derivative is y’ = 5x^4 + 3x^2 + 2x - 1.
Evaluate the integral of 1 / (x^2 - 4x + 4) with respect to x.
The integral evaluates to -1 / (x - 2) + C.
Determine the value of c that satisfies the Mean Value Theorem for the function f(x) = x^3 - 3x^2 + 2x on the interval [0, 2].
The value of c is 4/3.
Find the equation of the tangent plane to the surface z = x^2 + y^2 - 3x + 2y at the point (1, -1, 2).
The equation of the tangent plane is z = 2x + 2y + 1.
Calculate the second derivative of f(x) = 4e^(2x) - 3x^2.
The second derivative is f’‘(x) = 16e^(2x) - 6.
Find the derivative of y = (2x - 1)^3 e^x.
The derivative is y’ = (2x - 1)^3(e^x + 3e^x).
Evaluate the integral of (2x + 1) / (x^2 - 3x + 2) with respect to x.
The integral evaluates to ln|x - 1| - ln|x - 2| + C.
Determine the equation of the line normal to the curve y = 2x^3 + 3x - 1 at the point where x = -1.
The equation of the line is y = -11x + 3.
Find the absolute maximum and minimum values of the function f(x) = 4x^3 - 3x^2 - 6x + 2 on the interval [-1, 2].
The absolute maximum is f(2) = 16, and the absolute minimum is f(-1) = -3.
Evaluate the integral of cos^2(x) with respect to x.
The integral evaluates to (1/2)(x + sin(2x)) + C.
Determine the critical points of the function f(x) = x^4 - 4x^3 + 6x^2 - 8x.
The critical points are x = 0, x = 2.
Find the equation of the tangent line to the curve y = ln(x^2) at the point where x = e.
The equation of the tangent line is y = 2x - 1.
Calculate the second derivative of g(x) = 3x^2 - 2x + 1.
The second derivative is g’‘(x) = 6.
Evaluate the integral of (x^2 + 3x + 2) / (x^3 + x^2 + x) with respect to x.
The integral evaluates to (1/2) ln|x^2 + x| + 2 ln|x| - x + C.
Find the derivative of y = (x^2 + 1)^(3/2) + (2x^3 + 3x)^(4/3)
The derivative is y’ = 3x(x^2 + 1)^(1/2) + 4x^2(2x^3 + 3x)^1/3.
Consider the region bounded by the x-axis, the graph of y = 2x, and the line x = 2. Find the volume of the solid generated when this region is revolved around the x-axis.
The volume of the solid is 8π cubic units.
Given the curve y = x^3 and the x-axis, determine the volume of the solid formed when the region bounded by this curve and the lines x = 0 and x = 1 is rotated around the y-axis.
The volume of the solid formed when the region is rotated around the y-axis is (π/4) cubic units.
Find the volume of the solid obtained by revolving the region bounded by the curves y = x^2 and y = 2x around the line y = -1.
The volume of the solid is 56π/5 cubic units.
The region bounded by the curves y = e^x, y = 0, x = 0, and x = 1 is revolved around the y-axis. Calculate the volume of the resulting solid.
The volume of the solid is 2π(1 - 1/e) cubic units.
Consider the region bounded by the graph of y = x^2, the x-axis, and the lines x = 1 and x = 3. Find the volume of the solid formed when this region is revolved around the line y = -1.
The volume of the solid is 56π/5 cubic units.
