Exam 16: Trigonometric Models

Starting with the identity $\sin ^ { 2 } x + \cos ^ { 2 } x = 1$ and then dividing both sides of the equation by a suitable trigonometric function, derive the trigonometric identity. ​ $\sec ^ { 2 } x = 1 + \tan ^ { 2 } x$

Starting with the identity $\sin ^ { 2 } x + \cos ^ { 2 } x = 1$ , choose the right trigonometric identity.

Find the derivative of the function. $s ( x ) = 2 \cos ^ { 2 } x$

Evaluate the integral. $\int _ { 1 } ^ { \sqrt { \frac { \pi } { 2 } + 1 } } 4 x \cos \left( x ^ { 2 } - 1 \right) \mathrm { d } x$

Evaluate the integral. $\int - \sin ( - 4 x - 7 ) \mathrm { d } x$

Calculate the derivative. $\frac { d } { d x } \left[ \sec \left( \frac { 6 x ^ { 3 } } { x ^ { 2 } - 1 } \right) \right]$

Model the curve with a sine function. ​ ​ Note that the period of the curve is $P = \frac { 1 } { 4 }$ , its range is $[ - 1.9,1.9 ]$ and the graph of the sine function is shifted to the left 0.9 units. Write the model function as a function of x and π.

Model the curve with a cosine function. Note that the period of the curve is $P = \frac { 1 } { 3 }$ and its range is $[ - 1,1 ]$ .

Find the derivative of the function. $j ( x ) = 7 \sec ^ { 2 } x$

Calculate the derivative. ​ $\frac { d } { d x } \left[ e ^ { - 6 x } \sin ( 2 \pi x ) \right]$

Evaluate the integral. $\int _ { - \frac { \pi } { 4 } } ^ { \frac { \pi } { 4 } } \tan x \mathrm {~d} x$

Model the curve with a cosine function. Note that the period of the curve is $P = 14$ , its range is $[ 0,120 ]$ the graph of the cosine function is shifted upward 60 units and shifted to the right 7 units.

Decide whether each integral converges. If the integral converges, compute its value. Choose the correct letter for each question. -diverges

Calculate the derivative. $\frac { d } { d x } \left[ e ^ { - 3 x } \sin ( 3 \pi x ) \right]$

Recall that the average of a function $f ( x )$ on an interval $[ a , b ]$ is $\bar { f } = \frac { 1 } { b - a } \int _ { a } ^ { b } f ( x ) \mathrm { d } x$ Find the average of the given function. $f ( x ) = \sin ( 5 x )$ over $[ 0,5 \pi ]$

Model the curve with a cosine function. Note that the period of the curve is $P = \frac { 1 } { 5 }$ , its range is $[ - 3.3,3.3 ]$ and the graph of the cosine function is shifted to the right 0.35 units.

Sales of computers are subject to seasonal fluctuations. Computer City's sales of computers in 1995 and 1996 can be approximated by the function ​ $s ( t ) = 0.445 \sin ( 1.37 t + 1.75 ) + 0.712 ( 1 \leq t \leq 6 )$ ​ where ​t is time in quarters ( $t = 1$ represents the end of the first quarter of 1995) and $s ( t )$ is computer sales (quarterly revenue) in billions of dollars. Estimate Computer City's maximum and minimum quarterly revenue from computer sales. ​ Maximum sales __________ billions of dollars ​ Minimum sales __________ billions of dollars

Use the conversion formula $\cos x = \sin \left( \frac { \pi } { 2 } - x \right)$ to replace the expression $g ( x ) = 25 \cos [ 2 \pi ( 3 x - 1 ) ] + 9$ By a sine function.

Evaluate the integral. $\int \left( 3.4 \sec ^ { 2 } x + \frac { \cos x } { 1.4 } - 3.6 e ^ { x } \right) \mathrm { d } x$

Model the curve with a sine function. Note that the period of the curve is $P = \frac { 1 } { 5 }$ and its range is $[ - 2.2,2.2 ]$ and the graph of the sine function is shifted to the left 0.55 units.