Trigonometry Curve Sketching

11 Questions

Match the following functions with their derivatives:

$f(x) = x^2$ = $f'(x) = 2x$ $f(x) = 3x^2 + 2x$ = $f'(x) = 6x + 2$ $f(x) = 2x^3 - 5x$ = $f'(x) = 6x^2 - 5$ $f(x) = x^4 - 2x^2 + x$ = $f'(x) = 4x^3 - 4x + 1$

Match the following functions with their integrals:

$∫(2x + 1) dx$ = $x^2 + x + C$ $∫(x^2 - 3x + 2) dx$ = $(1/3)x^3 - (3/2)x^2 + 2x + C$ $∫(3x^2 - 2x + 1) dx$ = $x^3 - x^2 + x + C$ $∫(x^3 - 2x^2 + x) dx$ = $(1/4)x^4 - (2/3)x^3 + (1/2)x^2 + C$

Match the following functions with their stationary points:

$f(x) = x^2 - 4x + 3$ = $x = 1, x = 3$ $f(x) = x^3 - 6x^2 + 9x + 2$ = $x = 1, x = 3$ $f(x) = 2x^2 - 3x + 1$ = $x = 1/2, x = 1$ $f(x) = x^4 - 4x^3 + 7x^2 - 2x$ = $x = 0, x = 1, x = 2$

Match the following functions with their maximum or minimum values:

$f(x) = x^2 - 4x + 3$ = $f(2) = -1$ (minimum) $f(x) = x^3 - 6x^2 + 9x + 2$ = $f(1) = 0$ (minimum), $f(3) = 2$ (maximum) $f(x) = 2x^2 - 3x + 1$ = $f(3/4) = 1/8$ (minimum) $f(x) = x^4 - 4x^3 + 7x^2 - 2x$ = $f(1) = -2$ (minimum), $f(2) = 2$ (maximum)

Match the following functions with their gradient functions:

$f(x) = x^2 - 2x + 1$ = $f'(x) = 2x - 2$ $f(x) = x^3 - 3x^2 + 2x$ = $f'(x) = 3x^2 - 6x + 2$ $f(x) = 2x^2 - 3x + 2$ = $f'(x) = 4x - 3$ $f(x) = x^4 - 2x^3 + x^2 - x$ = $f'(x) = 4x^3 - 6x^2 + 2x - 1$

Match the following functions with their points of inflexion:

$f(x) = x^3 - 3x^2 + 2x$ = $x = 1$ $f(x) = x^4 - 4x^3 + 7x^2 - 2x$ = $x = 1, x = 2$ $f(x) = 2x^3 - 3x^2 + x$ = $x = 1/2$ $f(x) = x^5 - 5x^4 + 10x^3 - 5x^2 + x$ = $x = 1, x = 2, x = 3$

Match the following functions with their areas under the curves:

$y = x^2 - 2x + 1$ from $x = 0$ to $x = 2$ = $(4/3)$ $y = x^3 - 3x^2 + 2x$ from $x = 0$ to $x = 3$ = $9$ $y = 2x^2 - 3x + 1$ from $x = 0$ to $x = 2$ = $(8/3)$ $y = x^4 - 4x^3 + 7x^2 - 2x$ from $x = 0$ to $x = 4$ = $(32/5)$

Match the following functions with their volumes of revolution:

$y = x^2 - 2x + 1$ from $x = 0$ to $x = 2$ about the $x$-axis = $(16/3)π$ $y = x^3 - 3x^2 + 2x$ from $x = 0$ to $x = 3$ about the $x$-axis = $54π$ $y = 2x^2 - 3x + 1$ from $x = 0$ to $x = 2$ about the $x$-axis = $(24/3)π$ $y = x^4 - 4x^3 + 7x^2 - 2x$ from $x = 0$ to $x = 4$ about the $x$-axis = $(256/5)π$

Match the following functions with their surface areas of revolution:

$y = x^2 - 2x + 1$ from $x = 0$ to $x = 2$ about the $x$-axis = $(16/3)π + 4π$ $y = x^3 - 3x^2 + 2x$ from $x = 0$ to $x = 3$ about the $x$-axis = $54π + 12π$ $y = 2x^2 - 3x + 1$ from $x = 0$ to $x = 2$ about the $x$-axis = $(24/3)π + 6π$ $y = x^4 - 4x^3 + 7x^2 - 2x$ from $x = 0$ to $x = 4$ about the $x$-axis = $(256/5)π + 20π$

Match the following functions with their arc lengths:

$y = x^2 - 2x + 1$ from $x = 0$ to $x = 2$ = $(4 + 4√2)$ $y = x^3 - 3x^2 + 2x$ from $x = 0$ to $x = 3$ = $(9 + 9√3)$ $y = 2x^2 - 3x + 1$ from $x = 0$ to $x = 2$ = $(4 + 2√5)$ $y = x^4 - 4x^3 + 7x^2 - 2x$ from $x = 0$ to $x = 4$ = $(16 + 16√5)$

Match the following functions with their centre of curvature:

$y = x^2 - 2x + 1$ at $x = 1$ = $(-1, 2)$ $y = x^3 - 3x^2 + 2x$ at $x = 2$ = $(2, 4)$ $y = 2x^2 - 3x + 1$ at $x = 3/2$ = $(3/2, 5/2)$ $y = x^4 - 4x^3 + 7x^2 - 2x$ at $x = 2$ = $(2, 10)$

This quiz is about understanding the graphs of trigonometric functions, specifically cosine curves. It involves identifying coordinates of points on the curve and finding maximum/minimum values. Test your skills and knowledge of trigonometry!