3.5 Derivatives of trigonometric functions  (Page 3/3)

$y=x^2-\text{sec}x+1$

$y=3\text{csc}x+\frac{5}{x}$

$y=x^2\text{cot}x$

$y=x-x^3\text{sin}x$

$y=\frac{\text{sec}x}{x}$

$y=\text{sin}x\text{tan}x$

$y=\left(x+\text{cos}x\right)\left(1-\text{sin}x\right)$

$y=\frac{\text{tan}x}{1-\text{sec}x}$

$y=\frac{1-\text{cot}x}{1+\text{cot}x}$

$y=\text{cos}x\left(1+\text{csc}x\right)$

[T] $f\left(x\right)=\text{−}\text{sin}x,x=0$

[T] $f\left(x\right)=\text{csc}x,x=\frac{\pi }{2}$

[T] $f\left(x\right)=1+\text{cos}x,x=\frac{3\pi }{2}$

[T] $f\left(x\right)=\text{sec}x,x=\frac{\pi }{4}$

[T] $f\left(x\right)=x^2-\text{tan}xx=0$

[T] $f\left(x\right)=5\text{cot}xx=\frac{\pi }{4}$

$y=x\text{sin}x-\text{cos}x$

$y=\text{sin}x\text{cos}x$

$y=x-\frac{1}{2}\text{sin}x$

$y=\frac{1}{x}+\text{tan}x$

$y=2\text{csc}x$

$y={\text{sec}}^{2}x$

Find all $x$ values on the graph of $f\left(x\right)=\mathrm{-3}\text{sin}x\text{cos}x$ where the tangent line is horizontal.

Find all $x$ values on the graph of $f\left(x\right)=x-2\text{cos}x$ for $0<x<2\pi$ where the tangent line has slope 2.

Let $f\left(x\right)=\text{cot}x.$ Determine the points on the graph of $f$ for $0<x<2\pi$ where the tangent line(s) is (are) parallel to the line $y=\mathrm{-2}x.$

[T] A mass on a spring bounces up and down in simple harmonic motion, modeled by the function $s\left(t\right)=\mathrm{-6}\text{cos}t$ where $s$ is measured in inches and $t$ is measured in seconds. Find the rate at which the spring is oscillating at $t=5$ s.

Let the position of a swinging pendulum in simple harmonic motion be given by $s\left(t\right)=a\text{cos}t+b\text{sin}t.$ Find the constants $a$ and $b$ such that when the velocity is 3 cm/s, $s=0$ and $t=0.$

After a diver jumps off a diving board, the edge of the board oscillates with position given by $s(t)=\mathrm{-5}\text{cos}t$ cm at $t$ seconds after the jump.

1. Sketch one period of the position function for $t\ge 0.$
2. Find the velocity function.
3. Sketch one period of the velocity function for $t\ge 0.$
4. Determine the times when the velocity is 0 over one period.
5. Find the acceleration function.
6. Sketch one period of the acceleration function for $t\ge 0.$

The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by $y=10+5\text{sin}x$ where $y$ is the number of hamburgers sold and $x$ represents the number of hours after the restaurant opened at 11 a.m. until 11 p.m., when the store closes. Find $y\prime$ and determine the intervals where the number of burgers being sold is increasing.

[T] The amount of rainfall per month in Phoenix, Arizona, can be approximated by $y\left(t\right)=0.5+0.3\text{cos}t,$ where $t$ is months since January. Find $y^{\prime }$ and use a calculator to determine the intervals where the amount of rain falling is decreasing.

$\frac{d}{dx}(\text{cot}x)=\text{−}{\text{csc}}^{2}x$

$\frac{d}{dx}(\text{sec}x)=\text{sec}x\text{tan}x$

$\frac{d}{dx}(\text{csc}x)=\text{−}\text{csc}x\text{cot}x$

Use the definition of derivative and the identity

$\text{cos}\left(x+h\right)=\text{cos}x\text{cos}h-\text{sin}x\text{sin}h$ to prove that $\frac{d\left(\text{cos}x\right)}{dx}=\text{−}\text{sin}x.$

$\frac{d^{3}y}{d{x}^3}$ of $y=3\text{cos}x$

$\frac{d^{2}y}{d{x}^2}$ of $y=3\text{sin}x+x^2\text{cos}x$

$\frac{d^{4}y}{d{x}^4}$ of $y=5\text{cos}x$

$\frac{d^{2}y}{d{x}^2}$ of $y=\text{sec}x+\text{cot}x$

$\frac{d^{3}y}{d{x}^3}$ of $y=x^{10}-\text{sec}x$