EMath 1101 Exponential and Logarithmic Functions

False (B)

1/2

<p>-u' / √(1 - u²)</p> Signup and view all the answers

<p>2√(1 - x²) (C)</p> Signup and view all the answers

<p>arccos(u) = -u' / √(1 - u²) arctan(u) = u' / (1 + u²) arcsec(u) = u' / |u|√(u² - 1) arccsc(u) = -u' / |u|√(u² - 1)</p> Signup and view all the answers

<p>x = 0</p> Signup and view all the answers

<p>True (A)</p> Signup and view all the answers

<p>-π / 2 ≤ <em>y</em> ≤ π / 2 (B)</p> Signup and view all the answers

<p>True (A)</p> Signup and view all the answers

<p>e^x</p> Signup and view all the answers

<p>1</p> Signup and view all the answers

<p>sin(arcsin x) = x tan(arctan x) = x sec(arcsec x) = x arccos(cos y) = y</p> Signup and view all the answers

<p>0.305 (C)</p> Signup and view all the answers

<p>False (B)</p> Signup and view all the answers

<p>0 ≤ y ≤ π</p> Signup and view all the answers

<p><em>u’ / (1 + u^2)</em> (D)</p> Signup and view all the answers

<p>True (A)</p> Signup and view all the answers

<p>sinh(u) + C</p> Signup and view all the answers

<p>cosh(u)u'</p> Signup and view all the answers

<p><em>cosh^2(x) - sinh^2(x) = 1</em> (C)</p> Signup and view all the answers

<p>False (B)</p> Signup and view all the answers

<p>n*u^(n-1)*u'</p> Signup and view all the answers

<p>arcsin(u) = <em>u' / √(1 - u^2)</em> arccos(u) = <em>-u' / √(1 - u^2)</em> arctan(u) = <em>u' / (1 + u^2)</em> arccot(u) = <em>-u' / (1 + u^2)</em></p> Signup and view all the answers

<p><em>x</em>^<em>x</em> (1 + ln <em>x</em>) (B)</p> Signup and view all the answers

<p>True (A)</p> Signup and view all the answers

<p>π/3</p> Signup and view all the answers

<p>x</p> Signup and view all the answers

<p><em>y</em> = <em>arcsin x</em> = sin <em>y</em> = <em>x</em> <em>y</em> = <em>arccos x</em> = cos <em>y</em> = <em>x</em> <em>y</em> = <em>arctan x</em> = tan <em>y</em> = <em>x</em> <em>y</em> = <em>arcsec x</em> = sec <em>y</em> = <em>x</em></p> Signup and view all the answers

<p><em>tan(arctan x)</em> = x (B)</p> Signup and view all the answers

<p>False (B)</p> Signup and view all the answers

<p>-π/6</p> Signup and view all the answers

<p>$\frac{2}{\sqrt{1 - 4x^2}}$ (B)</p> Signup and view all the answers

<p>False (B)</p> Signup and view all the answers

<p>[-π/2, π/2]</p> Signup and view all the answers

<p>x = 0 (B)</p> Signup and view all the answers

<p>True (A)</p> Signup and view all the answers

<p>1</p> Signup and view all the answers

<p>sinh(u)u' (C)</p> Signup and view all the answers

<p>False (B)</p> Signup and view all the answers

<p>cos(u)u'</p> Signup and view all the answers

<p>-(csch(u) coth(u))u'</p> Signup and view all the answers

<p>sinh(u) = cosh(u)u' cosh(u) = sinh(u)u' tanh(u) = sech²(u)u' coth(u) = -(csch²(u))u'</p> Signup and view all the answers

<p>e^u * u' (D)</p> Signup and view all the answers

<p>False (B)</p> Signup and view all the answers

<p>u'/|u|, u ≠ 0</p> Signup and view all the answers

<p>(x - 1) sinh x + cosh^2 x (C)</p> Signup and view all the answers

<p>False (B)</p> Signup and view all the answers

<p>0 and 1</p> Signup and view all the answers

<p>concavity</p> Signup and view all the answers

<p><em>a^x</em> * <em>a^y</em> = <em>a^{x + y}</em> (B)</p> Signup and view all the answers

<p>False (B)</p> Signup and view all the answers

<p>0.30 grams</p> Signup and view all the answers

<p>1 / (ln(a) * x)</p> Signup and view all the answers

<p><em>a^0</em> = Always equals 1 <em>a^x</em> * <em>a^y</em> = Adds the exponents <em>a^x</em> / <em>a^y</em> = Subtracts the exponents (<em>a^x</em>)^y = Multiplies the exponents</p> Signup and view all the answers

<p>0.25 grams (D)</p> Signup and view all the answers

<p>True (A)</p> Signup and view all the answers

<p>0.5 grams</p> Signup and view all the answers

<p>Quotient Rule (C)</p> Signup and view all the answers

<p>True (A)</p> Signup and view all the answers

<p>-4</p> Signup and view all the answers

<p>exponential</p> Signup and view all the answers

<p>-sin x/(ln 10)cos x (A)</p> Signup and view all the answers

<p>log_a1 = 0 log_axy = log_ax + log_ay log_a(x/y) = log_ax - log_ay log_axⁿ = n log_ax</p> Signup and view all the answers

<p>nx^n-1</p> Signup and view all the answers

<p>True (A)</p> Signup and view all the answers

<p>0.5 grams (C)</p> Signup and view all the answers

<p>False (B)</p> Signup and view all the answers

<p>5715 years</p> Signup and view all the answers

<p>x - y</p> Signup and view all the answers

<p>a^0 = Equal to 1 a^x * a^y = Equal to a^(x+y) (a^x)^y = Equal to a^(xy) a^x / a^y = Equal to a^(x-y)</p> Signup and view all the answers

<p>0.30 grams (C)</p> Signup and view all the answers

<p>True (A)</p> Signup and view all the answers

<p>y = (1/2)^(t/5715)</p> Signup and view all the answers

<p>(ln 3)3^{x} (A), 3^{x} ln(3) (C)</p> Signup and view all the answers

<p>True (A)</p> Signup and view all the answers

<p>1/16</p> Signup and view all the answers

<p>x</p> Signup and view all the answers

<p>$log_{a}(1)$ = 0 $log_{a}(a^{x})$ = x $log_{a}(x/y)$ = $log_{a} x - log_{a} y$ $log_{a}(xy)$ = $log_{a} x + log_{a} y</p> Signup and view all the answers

<p>$log_{a} x = log_{b} x / log_{b} a$ (C)</p> Signup and view all the answers

<p>True (A)</p> Signup and view all the answers

<p>(3 ln 2)2^{3x}</p> Signup and view all the answers

<p>arctan(x): -∞ &lt; x &lt; ∞ (A), arcsin(x): -1 ≤ x ≤ 1 (B)</p> Signup and view all the answers

<p>True (A)</p> Signup and view all the answers

<p>π/6</p> Signup and view all the answers

<p>x^x(1 + ln x)</p> Signup and view all the answers

<p>arcsin(x) = -π/2 ≤ y ≤ π/2 arccos(x) = 0 ≤ y ≤ π arctan(x) = -π/2 &lt; y &lt; π/2 arcsec(x) = 0 ≤ y ≤ π, y ≠ π/2</p> Signup and view all the answers

<p>π (C)</p> Signup and view all the answers

<p>True (A)</p> Signup and view all the answers

<p>√(1 - (0.6)^2) = √(0.64) = 0.8</p> Signup and view all the answers

<p>u'v + uv' (A)</p> Signup and view all the answers

<p>False (B)</p> Signup and view all the answers

<p>e^u u'</p> Signup and view all the answers

<p>sinh(u) = cosh(u)u' cosh(u) = sinh(u)u' tanh(u) = sech²(u)u' sech(u) = -sech(u)tanh(u)u'</p> Signup and view all the answers

<p>False (B)</p> Signup and view all the answers

<p>sinh(u)u' (D)</p> Signup and view all the answers

<p>True (A)</p> Signup and view all the answers

<p>maximum, minimum</p> Signup and view all the answers

<p>u' / √(1 - u²)</p> Signup and view all the answers

<p>2 / √(1 - 4x²) (D)</p> Signup and view all the answers

<p>False (B)</p> Signup and view all the answers

<p>1/2</p> Signup and view all the answers

<p>2(1 - 2x arctan x)/(1 + x²)² (C)</p> Signup and view all the answers

<p>True (A)</p> Signup and view all the answers

<p>u' / |u|√(u² - 1)</p> Signup and view all the answers

The derivative of e^x with respect to x is e^x

The derivative of x^x with respect to x is xe^x-1

The inverse sine function; sin(arcsin x) = x

The inverse cosine function; cos(arccos x) = x

The inverse tangent function; tan(arctan x) = x

-π/6

π/2

π/3

The derivative of arcsin(u) with respect to x is u' / √(1 - u²).

The derivative of arccos(u) with respect to x is -u' / √(1 - u²).

The derivative of arctan(u) with respect to x is u' / (1 + u²).

The derivative of arcsec(u) with respect to x is u' / |u|√(u² - 1).

The process of finding the derivative of inverse trigonometric functions (arcsin, arccos, arctan).

The x-value that makes the first derivative equal to zero in an equation.

The horizontal line the graph approaches as x increases or decreases without bound.

The x values where the graph changes concavity.

The rate of change of the hyperbolic sine function sinh u with respect to x. Calculated as (cosh u) * u'.

The rate of change of the hyperbolic cosine function cosh u with respect to x. Calculated as (sinh u) * u'.

The rate of change of the hyperbolic tangent function tanh u with respect to x. Calculated as (sech² u) * u'.

The rate of change of the hyperbolic secant function sech u with respect to x. Calculated as - (sech u * tanh u) * u'.

The integral of the hyperbolic cosine function cosh u with respect to u. The result is sinh u + C, where C is the constant of integration.

The integral of the hyperbolic sine function sinh u with respect to u. The result is cosh u + C, where C is the constant of integration.

Functions that are related to the trigonometric functions but use hyperbolas instead of circles.

Used to differentiate composite functions; if y = f(u) and u = g(x), then dy/dx = df/du * du/dx*.

The logarithm of 1 to any base a is always equal to 0.

The logarithm of a product is equal to the sum of the logarithms of the individual factors.

The logarithm of a power is equal to the exponent multiplied by the logarithm of the base.

The logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator.

Two functions are inverses if, and only if, one function undoes the other's operation. This means that applying both functions in succession results in the original input value.

The derivative of log_ax with respect to x is 1 / (ln a)x.

The derivative of e^x with respect to x is simply e^x.

A function defined as a^x = e^{(ln a) x}, where a is a positive real number (a ≠ 1) and x is any real number.

The inverse of the exponential function to base a, defined as log_ax = 1 / ln(a) x, where a is a positive real number (a ≠ 1) and x is any positive real number.

A technique used to differentiate functions with complex exponents, like x^x, by taking the natural logarithm of both sides and applying properties of logarithms.

The time it takes for half of a radioactive substance to decay.

A mathematical model that describes the exponential decay of a radioactive substance using the half-life.

A method used to determine the age of organic materials by measuring the amount of carbon-14 remaining.

The usual laws of exponents apply to exponential functions with base a. For example:

• a⁰ = 1
• a^x a^y = a^{x + y}
• a^x / a^y = a^{x - y}
• (a^x)^y = a^xy

If -1 ≤ x ≤ 1, then sin(arcsin x) = x. This means the sine function 'undoes' the arcsine function.

If -∞ < x < ∞, then tan(arctan x) = x. This means the tangent function 'undoes' the arctangent function.

Use the half-life model: y = (½)^{(t/ 5715)}, where y is the amount of carbon-14, t is the time in years, and 5715 is the half-life of carbon-14.

If |x| ≥ 1, then sec(arcsec x) = x. This means the secant function 'undoes' the arcsecant function.

Logarithmic functions to the base a have similar properties to the natural logarithmic function.

The derivative of ln(cosh x) with respect to x is sinh x / cosh x, which simplifies to tanh x.

The derivative of x sinh x - cosh x with respect to x is x cosh x. You can apply the product rule and the derivative of sinh x and cosh x.

Relative extrema are the points on a function's graph where the function reaches a local maximum or minimum value. They are found by setting the first derivative equal to zero and solving for x.

A catenary is the curve formed by a hanging cable or chain suspended between two points. Its equation is given by y = a cosh(x/a), where a is the tension in the cable.

A critical number is a value in the domain of a function that makes the first derivative equal to zero or undefined.

A horizontal line that the graph of a function approaches as x approaches positive or negative infinity.

A point on a graph where the concavity changes from concave up to concave down, or vice-versa.

Finding the derivative of an inverse trigonometric function, like arcsin, arccos, arctan, and so on.

The derivative of a constant times a function is the constant times the derivative of the function.

The derivative of a sum or difference of functions is the sum or difference of their derivatives.

The derivative of the product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first.

The derivative of the quotient of two functions is equal to the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.

The derivative of x raised to a power n is n times x raised to the power n-1.

The derivative of the natural logarithm of a function u is the derivative of u divided by u itself.

The derivative of e raised to a function u is e raised to u multiplied by the derivative of u.

The derivative of the hyperbolic sine function sinh u is the hyperbolic cosine of u multiplied by the derivative of u.

The derivative of e^x with respect to x is simply e^x.

The logarithm of 1 to any base a is always equal to 0. In other words, a⁰ = 1.

The logarithm of a power is equal to the exponent multiplied by the logarithm of the base. In other words: log_axⁿ = n *log_a*x.

Finding the derivative of a function that is made up of two or more functions nested within each other. You need to apply the Chain Rule.

Used to differentiate composite functions. If y = f(u) and u = g(x), then dy/dx = df/du * du/dx*.