Differential Calculus: Growth and Derivatives

<p>It becomes inaccurate as growth slows down. (A)</p> Signup and view all the answers

<p>$rac{0.32}{0.15} = a^{10 - 5}$ (C)</p> Signup and view all the answers

<p>0.68 (B)</p> Signup and view all the answers

<p>u(t) = 0.07 · e^{0.148t} (C)</p> Signup and view all the answers

<p>Diminishing rate of growth as it ages. (C)</p> Signup and view all the answers

<p>0.022t + 1.1 (B)</p> Signup and view all the answers

<p>0.022 (A)</p> Signup and view all the answers

<p>By connecting two points with a secant line (A)</p> Signup and view all the answers

<p>The slope of the tangent line at a point (C)</p> Signup and view all the answers

<p>The slope is constant and equal to 2 (D)</p> Signup and view all the answers

<p>Calculate the limit of the difference quotient as Δx approaches 0 (A)</p> Signup and view all the answers

<p>u(t) = 0.038t + 0.7 (D)</p> Signup and view all the answers

<p>2 (A)</p> Signup and view all the answers

<p>The tangent line is identified at the point (D)</p> Signup and view all the answers

<p>u(t) = 0.07 * e^0.148t (A)</p> Signup and view all the answers

<p>The function is differentiable (C)</p> Signup and view all the answers

<p>It changes depending on the point (D)</p> Signup and view all the answers

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

<p>The rate of growth of the function (B)</p> Signup and view all the answers

<p>The product of the outer derivative and the inner derivative (C)</p> Signup and view all the answers

<p>The numerator and denominator can be any differentiable functions (D)</p> Signup and view all the answers

<p>Outer: 6(3x - 4)5, Inner: 3 (D)</p> Signup and view all the answers

<p>u'(t) = 0.07 · 0.148 · e^(0.148 · t) (A)</p> Signup and view all the answers

<p>The second derivative measures the slope of the first derivative (B)</p> Signup and view all the answers

<p>Product rule (D)</p> Signup and view all the answers

<p>f'(x) = g'(h) · h'(x) (C)</p> Signup and view all the answers

<p>Either chain rule or quotient rule (B)</p> Signup and view all the answers

<p>(ln a) · a^x (B)</p> Signup and view all the answers

<p>0.01036, 0.012 (A)</p> Signup and view all the answers

<p>The function nested within another function (C)</p> Signup and view all the answers

<p>u''(t) = 0.01036 · e^(0.148 · t)· 0.148 (C)</p> Signup and view all the answers

<p>It remains unchanged during differentiation (C)</p> Signup and view all the answers

<p>The degree decreases by 1 for each derivative. (A)</p> Signup and view all the answers

<p>An infinite sum of polynomial approximations around a point. (A)</p> Signup and view all the answers

<p>All of the above. (D)</p> Signup and view all the answers

<p>All derivatives equal the function itself, $e^x$. (C)</p> Signup and view all the answers

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

<p>A finite approximation of the Taylor series up to degree n. (A)</p> Signup and view all the answers

<p>They repeat every four derivatives. (B)</p> Signup and view all the answers

<p>The approximation becomes less accurate. (A)</p> Signup and view all the answers

<p>Only the constant term remains after the fourth derivative. (C)</p> Signup and view all the answers

<p>24 (A)</p> Signup and view all the answers

<p>The Taylor series of the exponential function converges for all real x. (D)</p> Signup and view all the answers

<p>Only odd powers of x appear with alternating signs. (B)</p> Signup and view all the answers

<p>Each coefficient equals the derivative at zero divided by factorial. (C)</p> Signup and view all the answers

<p>A constant value for all derivatives. (A)</p> Signup and view all the answers

<p>$6 imes 3 imes x^2$ (C), $18 imes x^2$ (D)</p> Signup and view all the answers

<p>It drops out. (C)</p> Signup and view all the answers

<p>$f_1' x imes f_2 x + f_1 x imes f_2' x$ (B)</p> Signup and view all the answers

<p>$f(x) = e^x$ (C)</p> Signup and view all the answers

<p>$rac{1}{2}x^{-rac{1}{2}}$ (C)</p> Signup and view all the answers

<p>$15x^4 + 3e^x - 3 imes rac{34}{x} - 2 imes sin(x)$ (B)</p> Signup and view all the answers

<p>$f'(x) = u'(x) + v'(x)$ (D)</p> Signup and view all the answers

<p>$f'(x) = rac{f_2(x)f_1'(x) - f_1(x)f_2'(x)}{f_2(x)^2}$ (A)</p> Signup and view all the answers

<p>$2x$ (D)</p> Signup and view all the answers

<p>Using the power rule. (A)</p> Signup and view all the answers

<p>$20x^4 + 6x^2$ (C)</p> Signup and view all the answers

<p>$cos(x)$ (B)</p> Signup and view all the answers

<p>0 (A)</p> Signup and view all the answers

<p>$rac{1}{x}$ (C)</p> Signup and view all the answers

<p>$f_1'(x) imes f_2(x) + f_1(x) imes f_2'(x)$ (A)</p> Signup and view all the answers

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

<p>(-2, 8) (A), (2, 8) (B)</p> Signup and view all the answers

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

<p>f′(x) has one zero (D)</p> Signup and view all the answers

<p>f(x) approaches zero from below (C)</p> Signup and view all the answers

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

<p>Wkin = mv^2 (B)</p> Signup and view all the answers

<p>It is strictly monotone decreasing for x &gt; -1 (A)</p> Signup and view all the answers

<p>The function does not have discontinuities (D)</p> Signup and view all the answers

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

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

<p>f: ℝ^2 → ℝ (C)</p> Signup and view all the answers

<p>Central symmetry (D)</p> Signup and view all the answers

<p>There is a local extreme value (C)</p> Signup and view all the answers

<p>$T_{sin} x = Σ_{k=0}^{∞} \frac{(-1)^k x^{2k + 1}}{(2k + 1)!}$ (B)</p> Signup and view all the answers

<p>$T_{sin,5} x = x - \frac{1}{6}x^3 + \frac{1}{120}x^5$ (B)</p> Signup and view all the answers

<p>f^{(4)}(x) (D)</p> Signup and view all the answers

<p>f'(x) &gt; 0 (B)</p> Signup and view all the answers

<p>Symmetric to the origin (B)</p> Signup and view all the answers

<p>f(x) has a turning point. (B)</p> Signup and view all the answers

<p>Only even powers of x (C)</p> Signup and view all the answers

<p>It converges for all real x. (A)</p> Signup and view all the answers

<p>Tends towards zero (C)</p> Signup and view all the answers

<p>The function value tends to infinity. (D)</p> Signup and view all the answers

<p>f'(x) &gt; 0 (D)</p> Signup and view all the answers

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

<p>It is defined for all real x. (B)</p> Signup and view all the answers

<p>The distance of a point from the origin in 2D space. (B)</p> Signup and view all the answers

<p>Direction affects the slope in a multidimensional function. (D)</p> Signup and view all the answers

<p>x2 is kept constant. (B)</p> Signup and view all the answers

<p>9 (D)</p> Signup and view all the answers

<p>The first-order partial derivative with respect to x2. (B)</p> Signup and view all the answers

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

<p>They provide information about the function's extremes. (D)</p> Signup and view all the answers

<p>It depends on the chosen direction of approach. (D)</p> Signup and view all the answers

<p>It indicates the rate of change in a particular direction. (B)</p> Signup and view all the answers

<p>They maintain symmetry among variables. (D)</p> Signup and view all the answers

<p>It involves combining multiple functions into one. (D)</p> Signup and view all the answers

<p>10 (D)</p> Signup and view all the answers

<p>They simplify complex functions into polynomial form. (C)</p> Signup and view all the answers

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

A method for finding the slope of a secant line—the line connecting two points on a curve.

The rate of change between two points on a curve.

A function of the form f(x) = a * b^x, where 'a' and 'b' are constants.

A process where a quantity increases over time at a constant proportional rate.

The process of finding the derivative of a function.

A function that describes the instantaneous rate of change of another function.

Derivatives of a derivative; successive rates of change.

Approximating a function with a polynomial function.

A function defined by multiple linear segments, each valid over a specific interval.

The rate of change of a linear function.

A line that intersects a curve at two points.

A line that touches a curve at only one point.

Another name for the instantaneous rate of change of a function at a point.

The instantaneous rate of change of a function, found using the differential quotient.

The function that gives the derivative values for all inputs.

A function with a unique, finite derivative at a point.

A rule for calculating the derivative of a power function (x^n).

The slope of a linear function is the coefficient of the x term.

The derivative of x^2 is 2x

A function formed from two or more functions, usually by substituting one into another.

The rate at which a function changes over time or with respect to an independent variable

Relates the tree's age to its circumference.

The derivative of a constant term is always zero.

To find the derivative of a sum of functions, differentiate each part of the sum individually and add the results.

To find the derivative of a product of two functions, differentiate each function individually and multiply the derivative of one function by the original other function. Then add these two products.

To find the derivative of a fraction of two functions, the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all divided by the squared denominator.

The instantaneous rate of change of a function at a particular point. It represents the slope of the tangent line at that point.

The notation used to represent the derivative of a function f(x). It is read as 'f prime of x'.

The derivative of sin(x) is cos(x).

The derivative of cos(x) is -sin(x).

The derivative of e^x is e^x.

The derivative of ln(x) is 1/x.

The derivative of a^x is ln(a) * a^x.

The derivative of x^(1/2) is 1/(2*sqrt(x)).

The derivative of a polynomial function is found by applying the power rule to each term of the polynomial.

To find the critical points of a function, set the derivative equal to zero and solve for x. These points represent potential maximums, minimums, or points of inflection.

The rate of change of the first derivative of a function.

The rate of change of the second derivative of a function.

The derivative of a function taken 'n' times.

Each derivative of a polynomial function decreases its degree by 1, until all derivatives are 0.

An infinite series representation of a function using its derivatives at a specific point.

A finite polynomial approximation of a function using its derivatives at a point.

The specific point around which we build the Taylor Series or Taylor Polynomial.

The Taylor Series of the exponential function e^x is a series with terms 1/k! * x^k, where k goes from 0 to infinity.

The Taylor Series of the sine function sin(x) is a series with terms x^(2k+1) / (2k+1)! * (-1)^k , where k goes from 0 to infinity.

Taylor series are used to approximate complicated functions with polynomials, which allows for easier calculations.

The Taylor Polynomial of e^x is an approximation of the function for a chosen number of terms.

The Taylor Polynomial of sin(x) is an approximation of the function for a chosen number of terms.

Calculators use Taylor series to approximate trigonometric functions and other complex functions for quick calculations.

The Taylor series of the sine function is an infinite sum that provides a perfect approximation of the sine function for all real numbers. The terms alternate in sign and involve odd powers of x.

The Taylor series of the cosine function is an infinite sum that provides a perfect approximation of the cosine function for all real numbers. The terms alternate in sign and involve even powers of x.

A point where a function is undefined, often leading to the function values approaching infinity in its vicinity.

Analyzing a function's behavior across its entire domain by examining its key features like symmetry, zeros, singularities, and asymptotic behavior.

The set of all possible input values for a function.

The set of all possible output values for a function.

A function has symmetry if its graph mirrors itself across a line or point.

An input value for a function that results in an output of zero. Where the function crosses the x-axis.

The behavior of a function as its input values approach positive or negative infinity, or a boundary of the domain.

A function is monotonic if it either consistently increases or consistently decreases over a certain interval.

A maximum or minimum point of a function within a given interval.

A point where the function changes its direction of concavity (from upwards to downwards or vice versa).

A point where the function has zero slope and its concavity changes in different directions - like a saddle.

A function that takes multiple input values and produces a single output value, e.g., f(x1, x2) = x1^2 + x2^2

The visual representation of a multivariable function in a 3D coordinate system, showing the output value for each input combination

The derivative of a multivariable function with respect to a single variable while holding all other variables constant

The notation for the partial derivative of function f with respect to variable x1

The rate of change of a multivariable function at a specific point, represented by a vector containing all partial derivatives

The derivative of a partial derivative, reflecting the rate of change of the slope in a certain direction

The notation for the second-order partial derivative of f, first with respect to x1 then x2

Second-order partial derivatives where the order of differentiation matters, e.g., ∂^2f/∂x1∂x2 and ∂^2f/∂x2∂x1

In many cases, mixed partial derivatives are equal regardless of the order of differentiation, e.g., ∂^2f/∂x1∂x2 = ∂^2f/∂x2∂x1

Maximum or minimum points of a multivariable function, where the gradient is zero or undefined

An infinite series representation of a multivariable function using partial derivatives, similar to the 1D Taylor series

Used to analyze and optimize problems involving multiple variables, like finding the maximum profit, minimizing costs, or determining the best design

A function that either always increases or always decreases over its entire domain. It never changes direction.

A point on a curve where the function's value is higher than all nearby points, but not necessarily the highest value overall.

A point on a curve where the function's value is lower than all nearby points, but not necessarily the lowest value overall.

A line that a curve approaches as its input values approach positive or negative infinity.

Set the first derivative of the function to zero and solve for x. Then check the sign of the second derivative at that x value. If the second derivative is negative, it's a local maximum. If it's positive, it's a local minimum.

Set the second derivative of the function to zero and solve for x. Make sure the third derivative is not zero at this point.

A rule for finding the derivative of a composite function. It's used when one function is inside another function.

A list of n numbers, representing a point in n-dimensional space. Example: (x1, x2, ..., xn)

Multivariable functions allow us to model real-world situations with multiple interconnected variables, enabling us to study complex relationships and dependencies between different factors.

The kinetic energy of a body depends on both its mass (m) and speed (v): Wkin(m, v) = (1/2) * m * v^2

In a composite function, the function that acts on the output of another function.

In a composite function, the function that is the input of another function.

The derivative of the outer function in a composite function.

The derivative of the inner function in a composite function.

The mathematical representation of the chain rule. The derivative of a composite function f(x) = g(h(x)) is equal to the derivative of the outer function g'(h) multiplied by the derivative of the inner function h'(x).

The outer function is g(h) = h^6, and the inner function is h(x) = 3x - 4. The outer derivative is g'(h) = 6h^5, and the inner derivative is h'(x) = 3. Applying the chain rule, f'(x) = g'(h) * h'(x) = 6(3x - 4)^5 * 3 = 18(3x - 4)^5

The derivative of a general exponential function f(x) = a^x is f'(x) = (ln a) * a^x. This can be derived using the chain rule and the definition of e as e^ln(a).

If u(t) represents the tree circumference at time t, then u'(t) represents the rate of change of the circumference with respect to time - how much the circumference grows per unit time.

The second derivative of the tree circumference function, u''(t), tells us how the rate of growth is changing over time. This indicates if the growth is accelerating or decelerating.

An alternative notation for representing the first derivative of a function f(x), read as 'df by dx'.

Differentiating a function f(x) means finding its first derivative, which gives the instantaneous rate of change at any point on the function's graph.