Calculus of Trigonometric Functions

NZ Level 8 (NZC) Level 3 (NCEA) [In development]

Basic derivatives of tan

Consider the graph of $y=\tan x$`y`=`t``a``n``x`.

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a

Which of the following best describes the graph of $y=\tan x$`y`=`t``a``n``x`?

The graph increases and decreases periodically.

A

It is constantly decreasing.

B

It is constantly increasing.

C

The graph increases and decreases periodically.

A

It is constantly decreasing.

B

It is constantly increasing.

C

b

Which of the following best describes the nature of the gradient of the curve?

Select all the correct options.

The gradient is always negative.

A

The gradient to the curve is never $0$0.

B

The gradient function has the same period as the curve itself.

C

The gradient increases more and more rapidly as the curve approaches the asymptotes.

D

The gradient is always positive.

E

The gradient to the curve is $0$0 every $\pi$π radians.

F

The gradient is always negative.

A

The gradient to the curve is never $0$0.

B

The gradient function has the same period as the curve itself.

C

The gradient increases more and more rapidly as the curve approaches the asymptotes.

D

The gradient is always positive.

E

The gradient to the curve is $0$0 every $\pi$π radians.

F

c

The tangent lines at the intercepts of the curve have been graphed as well.

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Using the graph, write down the gradient to the curve at $x=0$`x`=0, $\pi$π, $2\pi$2π, $3\pi$3π, $\text{. . .}$. . .

Gradient $=$= $\editable{}$.

d

The gradient function of $y=\tan x$`y`=`t``a``n``x` is $y'$`y`′. Which of the following is the correct graph of $y'$`y`′ for each value of $x$`x`?

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A

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B

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C

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D

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A

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B

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C

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D

e

Which of the following is the equation of the gradient function $y'$`y`′?

$y'=\sec^2\left(x\right)$`y`′=`s``e``c`2(`x`)

A

$y'=\sec x$`y`′=`s``e``c``x`

B

$y'=\csc^2\left(x\right)$`y`′=`c``s``c`2(`x`)

C

$y'=\csc x$`y`′=`c``s``c``x`

D

$y'=\sec^2\left(x\right)$`y`′=`s``e``c`2(`x`)

A

$y'=\sec x$`y`′=`s``e``c``x`

B

$y'=\csc^2\left(x\right)$`y`′=`c``s``c`2(`x`)

C

$y'=\csc x$`y`′=`c``s``c``x`

D

Easy

Approx 4 minutes

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