EMath 1101 Exponential and Logarithmic Functions

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Questions and Answers

What is the derivative of arcsin(u) with respect to x?

-u' / √(1 - u²)

u' / (1 - u²)

u' / √(1 - u²) (correct)

u' / √(1 + u²)

The function y = arctan(x) has a horizontal asymptote at y = π/2.

False

What is the value of tan(y) when y = arcsec(√5 / 2)?

1/2

The derivative of arccos(u) is _____.

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

Which of the following is the correct simplification of the derivative y' = arcsin(x) + x√(1 - x²)?

2√(1 - x²) Signup and view all the answers

Match the inverse trigonometric functions with their derivatives:

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

What are the critical numbers for the function y = (arctan x)²?

x = 0 Signup and view all the answers

The graph of y = (arccos x) has a restricted domain of [-1, 1].

True Signup and view all the answers

What is the correct range for the function y = arcsin x?

-π / 2 ≤ y ≤ π / 2 Signup and view all the answers

The value of arccos(0) is π/2.

True Signup and view all the answers

What is the derivative of y = e^x?

e^x Signup and view all the answers

The equation arctan(2x - 3) = π/4 can be solved by finding that 2x - 3 equals ________.

1 Signup and view all the answers

Match the following inverse trigonometric identities with their corresponding properties:

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

Given y = arcsin(0.3), what is the approximate value of y?

0.305 Signup and view all the answers

If y = arcsec(√5/2), then tan y equals 1.

False Signup and view all the answers

If the domain of y = arccos x is -1 ≤ x ≤ 1, then the range of y is ________.

0 ≤ y ≤ π Signup and view all the answers

Which of the following is the derivative of the function arctan(u)?

u’ / (1 + u^2) Signup and view all the answers

The derivative of arcsec(u) is given by u’ / |u|√(u^2 - 1).

True Signup and view all the answers

What is the integral of cosh(u)?

sinh(u) + C Signup and view all the answers

The derivative of the function sinh(u) is __________.

cosh(u)u' Signup and view all the answers

Which hyperbolic identity is correct?

cosh^2(x) - sinh^2(x) = 1 Signup and view all the answers

The derivative of sec(u) includes a negative sign.

False Signup and view all the answers

What is the result of differentiating the function u^n?

n*u^(n-1)*u' Signup and view all the answers

Match the following inverse trig functions with their derivatives:

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

What is the derivative of the function y = x^x?

xx (1 + ln x) Signup and view all the answers

The function y = arccos x has a range of [0, π].

True Signup and view all the answers

What is the value of arctan(√3)?

π/3 Signup and view all the answers

For the function y = arcsin x, the relationship sin(y) = _____ holds true.

x Signup and view all the answers

Match the following inverse trigonometric functions with their definitions:

y = arcsin x = sin y = x y = arccos x = cos y = x y = arctan x = tan y = x y = arcsec x = sec y = x Signup and view all the answers

Which property holds true for y = arctan x?

tan(arctan x) = x Signup and view all the answers

y = arccsc x is defined for all real numbers.

False Signup and view all the answers

The value of arcsin(-½) is approximately _____ radians.

-π/6 Signup and view all the answers

What is the derivative of the function y = arcsin(2x)?

$\frac{2}{\sqrt{1 - 4x^2}}$ Signup and view all the answers

The derivative of arcsec u is negative.

False Signup and view all the answers

The range of the function y = arcsin x is ________.

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

In analyzing the graph of y = (arctan x)², which of the following points indicates a relative minimum?

x = 0 Signup and view all the answers

The horizontal asymptote of y = (arctan x)² is π² / 4.

True Signup and view all the answers

For y = arcsec(√5/2), what is tan y?

1 Signup and view all the answers

What is the derivative of the function y = cosh(u) with respect to x?

sinh(u)u' Signup and view all the answers

The identity tanh²(x) + sech²(x) = 1 is true.

False Signup and view all the answers

State the derivative of sin(u).

cos(u)u' Signup and view all the answers

The derivative of csch(u) is __________.

-(csch(u) coth(u))u' Signup and view all the answers

Match the following hyperbolic functions with their derivatives:

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

Which of the following formulas represents the chain rule for the differentiation of e^(u)?

e^u * u' Signup and view all the answers

The derivative of a constant function is always 1.

False Signup and view all the answers

What is the result of differentiating |u|?

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

What is the first derivative of the function f(x) = (x - 1) cosh x - sinh x?

(x - 1) sinh x + cosh^2 x Signup and view all the answers

The equation for the shape of a hanging cable is represented by y = a sinh(x/a).

False Signup and view all the answers

What are the critical numbers for the function f(x) = (x - 1) cosh x - sinh x?

0 and 1 Signup and view all the answers

The second derivative test helps identify the nature of critical points by providing information about the ______.

concavity Signup and view all the answers

Which of the following statements is true about the exponential function to the base a?

a^x * a^y = a^{x + y} Signup and view all the answers

The function y = ½^(t / 5715) represents a growth model for carbon-14.

False Signup and view all the answers

What is the approximate amount of carbon-14 remaining after 10,000 years if the half-life is 5715 years?

0.30 grams Signup and view all the answers

The logarithmic function to base a is defined as log_a(x) = _____.

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

Match the following properties of exponents with their descriptions:

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

If a sample contains 1 gram of carbon-14, how much will remain after 2 half-lives?

0.25 grams Signup and view all the answers

The statement log_a(xy) = log_a(x) + log_a(y) represents a property of logarithms.

True Signup and view all the answers

What is the value of y when t equals 5715 in the carbon-14 decay model?

0.5 grams Signup and view all the answers

Which of the following properties of logarithms states that log_a(x/y) equals log_ax minus log_ay?

Quotient Rule Signup and view all the answers

The derivative of the function a^x is equal to (ln a)a^x for any positive real number a.

True Signup and view all the answers

What value does log₃1/81 yield?

-4 Signup and view all the answers

The logarithmic function is the inverse of the _____ function.

exponential Signup and view all the answers

What is the derivative of log₁₀cos x?

-sin x/(ln 10)cos x Signup and view all the answers

Match the following logarithmic identities with their expressions:

loga1 = 0 logaxy = logax + logay loga(x/y) = logax - logay logaxn = n logax Signup and view all the answers

What is the result of the derivative d/dx[xⁿ]?

nxn-1 Signup and view all the answers

For any positive base a, a^log_ax equals x for x > 0.

True Signup and view all the answers

What is the result of evaluating the function $y = (1/2)^{(t/5715)}$ when $t = 5715$?

0.5 grams Signup and view all the answers

The logarithmic function to a positive base a is defined as $log_a x = 1 / (ln a imes x)$.

False Signup and view all the answers

What is the half-life of carbon-14?

5715 years Signup and view all the answers

The property that states a^x / a^y = a^ is known as the property of exponents.

x - y Signup and view all the answers

Match the following exponential properties with their descriptions:

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) Signup and view all the answers

Given a sample of carbon-14 that starts at 1 gram, how much will remain after 10,000 years?

0.30 grams Signup and view all the answers

The function a^x is defined for any positive real number a that is not equal to 1.

True Signup and view all the answers

What is the formula to express exponential growth when using a base of ½?

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

What is the derivative of the function $y = 3^{x}$?

(ln 3)3^{x} Signup and view all the answers

The logarithmic identity $log_{a}(xy) = log_{a}(x) + log_{a}(y)$ is true.

True Signup and view all the answers

What is the value of $x$ in the equation $log_{2} x = -4$?

1/16 Signup and view all the answers

The property $log_{a}(y) = x$ implies that $y = a^{______}$.

x Signup and view all the answers

Match the logarithmic property with its description:

$log_{a}(1)$ = 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 Signup and view all the answers

Which of the following illustrates the base conversion for logarithms?

$log_{a} x = log_{b} x / log_{b} a$ Signup and view all the answers

The derivative of the natural logarithm function $log_{e}(x)$ is equal to $1/x$.

True Signup and view all the answers

Differentiate the function $y = 2^{3x}$ with respect to $x$.

(3 ln 2)2^{3x} Signup and view all the answers

Which of the following correctly matches the inverse trigonometric functions with their domains?

arctan(x): -∞ < x < ∞ Signup and view all the answers

The value of arcsin(1) is π/2.

True Signup and view all the answers

Evaluate arcsin(0.5).

π/6 Signup and view all the answers

The derivative of y = x^x is __________.

x^x(1 + ln x) Signup and view all the answers

Match each inverse trigonometric function with its corresponding range:

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

What is the value of arccos(-1)?

π Signup and view all the answers

Arctan(0) is equal to 0.

True Signup and view all the answers

Using right triangles, if y = arcsin(0.6), find cos(y) given that 0 < y < π/2.

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

What is the formula for the derivative of a product of two functions, u and v?

u'v + uv' Signup and view all the answers

The identity tanh²(x) + sech²(x) = 1 is correct.

False Signup and view all the answers

The derivative of the function y = e^u is ________.

e^u u' Signup and view all the answers

Match the hyperbolic function with its derivative:

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

The function y = a cosh(x/a) describes the shape of a hanging cable and is classified as a parabola.

False Signup and view all the answers

What is the derivative of cosh(u)?

sinh(u)u' Signup and view all the answers

Csc(u) has a negative derivative.

True Signup and view all the answers

The second derivative test is used to determine whether a critical point is a or a .

maximum, minimum Signup and view all the answers

What is the relationship between the derivatives of the arcsin function?

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

For the function y = arcsin(2x), what is the derivative dy/dx?

2 / √(1 - 4x²) Signup and view all the answers

The derivative of arcsec(u) is positive for all u when u > 1.

False Signup and view all the answers

What is the value of tan(y) when y = arcsec(√5/2)?

1/2 Signup and view all the answers

Which of the following represents the correct expression for the second derivative of y = (arctan x)²?

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

The critical number of the function y = (arctan x)² is at x = 0.

True Signup and view all the answers

What is the formula for the derivative of y = arcsec(u)?

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

Study Notes

EMath 1101 Study Notes

Exponential Function to Base a: If a is a positive real number (a ≠ 1) and x is any real number, the exponential function to base a is defined as a^x = e^{(ln a)x}. If a = 1, then y = 1^x = 1 (a constant function). These functions follow the usual laws of exponents.
Familiar Properties (Exponential Functions to Base a):
- a⁰ = 1
- a^xa^y = a^x+y
- a^x/a^y = a^x-y
- (a^x)^y = a^xy
Radioactive Half-Life Model (Example): Carbon-14 has a half-life of approximately 5715 years. An equation can model the amount of carbon-14 remaining over time (y = (1/2)^t/5715).
Logarithmic Function to Base a: If a is a positive real number (a ≠ 1) and x is any positive real number, the logarithmic function to base a is defined as log_ax = (1/ln a) * ln x.
Properties of Logarithmic Functions to Base a:
- log_a 1 = 0
- log_a(xy) = log_ax + log_ay
- log_a(xⁿ) = n log_ax
- log_a (x/y) = log_a x - log_a y
Exponential and Logarithmic Functions (Inverse Functions): f(x) = a^x and g(x) = log_ax are inverse functions of each other.
Properties of Inverse Functions:
- y = a^x if and only if x = log_ay
- a^log_ax = x (for x > 0)
- log_a(a^x) = x (for all x)
Common Logarithms: The logarithmic function to base 10 is called the common logarithm, denoted as log₁₀x or simply log x.
Derivatives for Bases Other Than e: Rules for differentiating exponential and logarithmic functions with bases other than e are provided involving natural logarithms (ln). These rules are extensions for chain rule.
The Power Rule for Real Exponents: For differentiable function u of x, and real number n
- D(xⁿ)/dx = nx^n-1
- D(uⁿ)/dx = nu^n-1 (du/dx)
Derivatives of Inverse Trigonometric Functions: Formulas are given for differentiating various inverse trigonometric functions like arcsin, arccos, arctan, etc., if u(x) is a differentiable function.
Solving Equations (Bases other than e): Examples demonstrate how to solve equations involving exponential and logarithmic functions using functions with base a.
Evaluating Inverse Trigonometric Functions: Examples show evaluation of inverse trigonometric functions using correct intervals for output values.
Properties of Inverse Trigonometric Functions: Relationships between sine, cosine, tangent and their inverses are detailed.
Differentiation of Inverse Trigonometric Functions (Examples): Specific examples showcase the application of derivative formulas to functions with inverse trigonometric functions.
Analyzing Inverse Trigonometric Graphs: Analysis of graphs for y = (arctan x)² is performed including finding asymptotes, inflection points, and critical points by differentiation.
Hyperbolic Functions: Identities for hyperbolic functions are presented. These include hyperbolic sine (sinh), cosine (cosh), tangent (tanh), and their inverses.
Derivatives and Integrals of Hyperbolic Functions: Formulas for differentiating and integrating hyperbolic functions are detailed.

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This quiz explores the properties and applications of exponential and logarithmic functions as covered in EMath 1101. It includes definitions, laws, and examples such as carbon-14's half-life model. Test your understanding of these fundamental mathematical concepts with this quiz.

EMath 1101 Exponential and Logarithmic Functions

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Questions and Answers

What is the derivative of arcsin(u) with respect to x?

The function y = arctan(x) has a horizontal asymptote at y = π/2.

What is the value of tan(y) when y = arcsec(√5 / 2)?

The derivative of arccos(u) is _____.

Which of the following is the correct simplification of the derivative y' = arcsin(x) + x√(1 - x²)?

Match the inverse trigonometric functions with their derivatives:

What are the critical numbers for the function y = (arctan x)²?

The graph of y = (arccos x) has a restricted domain of [-1, 1].

What is the correct range for the function y = arcsin x?

The value of arccos(0) is π/2.

What is the derivative of y = ex?

The equation arctan(2x - 3) = π/4 can be solved by finding that 2x - 3 equals ________.

Match the following inverse trigonometric identities with their corresponding properties:

Given y = arcsin(0.3), what is the approximate value of y?

If y = arcsec(√5/2), then tan y equals 1.

If the domain of y = arccos x is -1 ≤ x ≤ 1, then the range of y is ________.

Which of the following is the derivative of the function arctan(u)?

The derivative of arcsec(u) is given by u’ / |u|√(u^2 - 1).

What is the integral of cosh(u)?

The derivative of the function sinh(u) is __________.

Which hyperbolic identity is correct?

The derivative of sec(u) includes a negative sign.

What is the result of differentiating the function u^n?

Match the following inverse trig functions with their derivatives:

What is the derivative of the function y = xx?

The function y = arccos x has a range of [0, π].

What is the value of arctan(√3)?

For the function y = arcsin x, the relationship sin(y) = _____ holds true.

Match the following inverse trigonometric functions with their definitions:

Which property holds true for y = arctan x?

y = arccsc x is defined for all real numbers.

The value of arcsin(-½) is approximately _____ radians.

What is the derivative of the function y = arcsin(2x)?

The derivative of arcsec u is negative.

The range of the function y = arcsin x is ________.

In analyzing the graph of y = (arctan x)², which of the following points indicates a relative minimum?

The horizontal asymptote of y = (arctan x)² is π² / 4.

For y = arcsec(√5/2), what is tan y?

What is the derivative of the function y = cosh(u) with respect to x?

The identity tanh²(x) + sech²(x) = 1 is true.

State the derivative of sin(u).

The derivative of csch(u) is __________.

Match the following hyperbolic functions with their derivatives:

Which of the following formulas represents the chain rule for the differentiation of e^(u)?

The derivative of a constant function is always 1.

What is the result of differentiating |u|?

What is the first derivative of the function f(x) = (x - 1) cosh x - sinh x?

The equation for the shape of a hanging cable is represented by y = a sinh(x/a).

What are the critical numbers for the function f(x) = (x - 1) cosh x - sinh x?

The second derivative test helps identify the nature of critical points by providing information about the ______.

Which of the following statements is true about the exponential function to the base a?

The function y = ½^(t / 5715) represents a growth model for carbon-14.

What is the approximate amount of carbon-14 remaining after 10,000 years if the half-life is 5715 years?

The logarithmic function to base a is defined as log_a(x) = _____.

Match the following properties of exponents with their descriptions:

If a sample contains 1 gram of carbon-14, how much will remain after 2 half-lives?

The statement log_a(xy) = log_a(x) + log_a(y) represents a property of logarithms.

What is the value of y when t equals 5715 in the carbon-14 decay model?

Which of the following properties of logarithms states that loga(x/y) equals logax minus logay?

The derivative of the function ax is equal to (ln a)ax for any positive real number a.

What value does log31/81 yield?

The logarithmic function is the inverse of the _____ function.

What is the derivative of log10cos x?

Match the following logarithmic identities with their expressions:

What is the result of the derivative d/dx[xn]?

For any positive base a, alogax equals x for x > 0.

What is the result of evaluating the function $y = (1/2)^{(t/5715)}$ when $t = 5715$?

The logarithmic function to a positive base a is defined as $log_a x = 1 / (ln a imes x)$.

What is the half-life of carbon-14?

The property that states ax / ay = a______ is known as the ______ property of exponents.

Match the following exponential properties with their descriptions:

Given a sample of carbon-14 that starts at 1 gram, how much will remain after 10,000 years?

The function ax is defined for any positive real number a that is not equal to 1.

What is the formula to express exponential growth when using a base of ½?

What is the derivative of the function $y = 3^{x}$?

The logarithmic identity $log_{a}(xy) = log_{a}(x) + log_{a}(y)$ is true.

What is the derivative of y = e^x?

What is the derivative of the function y = x^x?

Which of the following properties of logarithms states that log_a(x/y) equals log_ax minus log_ay?

The derivative of the function a^x is equal to (ln a)a^x for any positive real number a.

What value does log₃1/81 yield?

What is the derivative of log₁₀cos x?

What is the result of the derivative d/dx[xⁿ]?

For any positive base a, a^log_ax equals x for x > 0.

The property that states a^x / a^y = a^ is known as the property of exponents.

The function a^x is defined for any positive real number a that is not equal to 1.

The second derivative test is used to determine whether a critical point is a or a .