⧼exchistory⧽
13 exercise(s) shown, 0 hidden
BBy Bot
Nov 03'24
[math]
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[/math]
Which of the six trigonometric functions are odd functions and which are even functions?
BBy Bot
Nov 03'24
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[/math]
Derive the formulas for the derivatives of the functions [math]\cot[/math], [math]\sec[/math], and [math]\csc[/math].
BBy Bot
Nov 03'24
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[/math]
Find the following derivatives.
- [a [math]\ddx \sec^2 x[/math]
- [math]\ddx \tan (2x^2 - 1)[/math]
- [math]\ddx \ln |\sec x|[/math]
- [math]\frac{d}{dy} \cos y \tan y[/math]
- [math]\frac{d}{dt} (\sec^2t - 1)[/math]
- [math]\ddx \csc (x^3 - 1)[/math]
- [math]\ddx \tan x \cot x[/math]
- [math]\frac{d}{dt} \ln |\cot t|[/math].
BBy Bot
Nov 03'24
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[/math]
Prove each of the following identities from the basic identities in sine and cosine developed in Section \secref{6.1}.
- [math]\tan(x+y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}[/math]
- [math]\csc x = \sec \left( x - \frac{\pi}2 \right)[/math]
- [math]\cot(a+b) = \frac{\cot a\cot b-1}{\cot a+\cot b}[/math]
- [math]\cot(x+\pi) = \cot x[/math].
BBy Bot
Nov 03'24
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[/math]
Find the following intervals.
- [math]\int \tan 5x \; dx[/math]
- [math]\int \cot x \; dx[/math]
- [math]\int e^x \sec^2 e^x \; dx[/math]
- [math]\int \tan^2 x \; dx[/math] \quad
- [math]\int \tan^4 x \sec^2 x \; dx[/math]
- [math]\int \sec^4 x \tan x \; dx[/math]
- [math]\int \frac1{\sqrt{x}} \csc \sqrt{x} \cot \sqrt{x} \; dx[/math]
- [math]\int \csc^4 x \; dx[/math].
BBy Bot
Nov 03'24
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[/math]
Find [math]\ddx (\sec x + \tan x)[/math] and use the result to evaluate the integral
[[math]]
\int \sec x \; dx = \int \sec x
\frac{\sec x + \tan x}{\sec x + \tan x} \; dx
.
[[/math]]
BBy Bot
Nov 03'24
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[/math]
Find [math]\int \tan x \; dx = \int \frac{\tan x \sec x}{\sec x} \; dx[/math] by substituting [math]u = \sec x[/math] and [math]\nxder{}ux[/math] in the right side. Compare the answer obtained with Example \ref{ex6.3.1b}.
BBy Bot
Nov 03'24
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[/math]
Draw the graph of
- [math]\cot x[/math]
- [math]\csc x[/math].
BBy Bot
Nov 03'24
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[/math]
Evaluate the following limits.
- [math]\lim_{x\goesto0} \frac{\tan x}x[/math]
- [math]\lim_{x\goesto0} x \cot 2x[/math].
BBy Bot
Nov 03'24
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[/math]
Find the tangent of the angle between
- the straight lines [math]y-2x=1[/math] and [math]2y-x=4[/math].
- the straight lines [math]y+2x=2[/math] and [math]2y-x=2[/math].
- the tangent lines to the curves [math]y=x^2[/math] and [math]x^2+y^2=1[/math] at their point of intersection in the first quadrant.