Free Printable Unit 2: Differentiation Vocab

1.

Shall we stand and fight?

a)

No, we should retreat instead.

b)

Let's negotiate for peace.

c)

We should run away and hide.

d)

Yes, we shall stand and fight.

2.

What is the slope intercept equation?

a)

y = b + mx

b)

y = mx + b

c)

y = mx - b

d)

y = m + bx

3.

What is the difference between instantaneous and average slope in a graph?

a)

The average slope is always steeper than the instantaneous slope.

b)

The instantaneous slope is calculated over a time interval, while the average slope is at a single point.

c)

The average slope is only relevant for linear graphs, while the instantaneous slope applies to all graphs.

d)

The average slope is the overall rate of change over an interval, while the instantaneous slope is the rate of change at a specific point.

4.

What is secant line?

a)

A secant line is a line that touches a curve at one point.

b)

A secant line is a line that runs parallel to a curve.

c)

A secant line is a line that intersects a curve at three or more points.

d)

A secant line is a line that intersects a curve at two or more points.

5.

What is a tangent line?

a)

A tangent line is a line that intersects a curve at two points.

b)

A tangent line is a line that is parallel to the curve at all points.

c)

A tangent line is a line that curves along with the curve it touches.

d)

A tangent line is a line that touches a curve at one point.

6.

What is a derivative?

a)

A derivative is a fixed value of a function.

b)

A derivative is a measure of how a function changes as its input changes.

c)

A derivative is a type of integral.

d)

A derivative measures the area under a curve.

7.

$\frac{\left(dy\right)}{\left(dx\right)}and\ \frac{\Delta y}{\Delta x}$ are essentially the same concept

a)

False

b)

True

8.

When can find the the slope of two points and find the average slope by....

a)

Use the slope formula and average the results.

b)

Average the x-coordinates of the points.

c)

Use the distance formula to find the slope.

d)

Calculate the slope using only one point.

9.

Lagrange Notation? When is it used? f'

a)

Lagrange notation is used to denote derivatives of functions in calculus.

b)

Lagrange notation is used to represent statistical data in research.

c)

Lagrange notation is a method for solving algebraic equations.

d)

Lagrange notation is used to calculate integrals in geometry.

10.

Lagrange Notation? When is it used? What is the symbols?

a)

Lagrange notation is used for integration, using symbols like ∫f(x)dx.

b)

Lagrange notation denotes limits, using symbols like lim f(x) as x approaches a.

c)

Lagrange notation is used for summation, using symbols like Σf(n) for series.

d)

Lagrange notation is used to denote derivatives, using symbols like f'(x) for the first derivative.

11.

Leibriz Notation? When is it used? What is the symbol?

a)

The symbol for Leibniz notation is ∫f(x)dx.

b)

The symbol for Leibniz notation is d²y/dx².

c)

The symbol for Leibniz notation is dy/dx.

d)

Leibniz notation is used for matrix operations.

12.

Newton Notions? When is it used? What is the symbol?

a)

Newton's laws of motion; symbol: F (force) in Newtons (N).

b)

Einstein's theory of relativity; symbol: E (energy) in Joules (J).

c)

Newton's law of universal gravitation; symbol: G (gravity) in meters (m).

d)

Newton's first law of thermodynamics; symbol: T (temperature) in Celsius (°C).

13.

Why is representing a dervitive like d/dx (x^) more better than than dy/dx, y=x^2

a)

d/dx (x) is more complex and less intuitive.

b)

Representing a derivative as d/dx (x^) is clearer and emphasizes the differentiation process.

c)

d/dx (x^2) is more accurate than d/dx (x^)

d)

dy/dx is universally accepted and easier to understand.

14.

How does the limit work with derivatives

a)

Derivatives are calculated without any reference to limits.

b)

The limit defines the derivative as the instantaneous rate of change of a function.

c)

The limit is irrelevant to the concept of derivatives.

d)

The limit only applies to integrals, not derivatives.

15.

$f'\left(x\right)=\lim_{h\rightarrow0}\frac{\left(f\left(x_o+h\right)-f\left(x_o\right)\right)}{h}$

What is an equation called?

a)

Formal Derivative

b)

Limit definition

c)

Derivative formula

d)

Slope equation

16.

$f'\left(x\right)=\lim_{h\rightarrow a}\ \ \frac{\left(\left(f\left(x\right)-f\left(a\right)\right)\right)}{x-a}$

What is this called?

a)

Function

b)

Limit

c)

Derivative

d)

Integral

17.

When deriving an equation, its quite simple. Use the dervitive formula and plug in in values for the initial and finial value with the hypotical additional points. This than allows you to find the derivative!

a)

Ignore the initial and final values when calculating the derivative.

b)

Substitute values without using the derivative formula.

c)

Use the integral formula to find the derivative.

d)

Use the derivative formula and substitute values to find the derivative.

18.

True or False: The derivative formula works for lots of equations

a)

True

b)

False

19.

Its important to analyze a limit of a derivative before making assumptions that a derivative is true.

a)

Derivatives do not require limit analysis before use.

b)

Limits of derivatives are irrelevant to their validity.

c)

Assuming a derivative is always valid is sufficient.

d)

Analyzing the limit of a derivative is crucial before assuming its validity.

20.

If a limit does not exist, what does that say about the derivative?

a)

The derivative does not exist.

b)

The derivative is always defined.

c)

The limit must be zero.

d)

The derivative is infinite.

21.

A function f(x) is called differentiable at x=a if f′(a) exists and f(x) is called differentiable on an interval if the derivative exists for each point in that interval.

a)

A function is differentiable at a point if its derivative exists at that point.

b)

A function is differentiable at a point if it has a maximum or minimum at that point.

c)

A function is differentiable on an interval if it is linear throughout that interval.

d)

A function is differentiable at a point if it is continuous at that point.

22.

If f(x) is NOT differentiable at x=a then f(x) is continuous at x=ax=a.

a)

True

b)

False

23.

$f′(x)=y′=\frac{\left(df\right)}{\left(dx\right)}=\frac{dy}{dx}=\frac{d}{dx}(f(x))=\frac{d}{dx}(y)\left(x=a\right)$

What does this represent.

a)

The integral of a function.

b)

The limit of a function.

c)

The derivative of a function.

d)

The average value of a function.

24.

Its important to know the definition of the derivative! It is an important definition that we should always know and keep in the back of our minds. It is just something that we’re not going to be working with all that much

a)

The derivative is the slope of a line at a point.

b)

The derivative is the average value of a function over an interval.

c)

The derivative measures the total change of a function.

d)

The derivative is the limit of the average rate of change of a function.

25.

True or False: The Larger the number the faster the rate of change. In terms of derivatives, the sign does not matter.

a)

True

b)

False

26.

This is the next major interpretation of the derivative. The slope of the tangent line to f(x) at x=a is f′(a). The tangent line then is given by,

a)

y = f(a) + f′(a)(x - a)

b)

y = f(a) + f(x)(x - a)

c)

y = f(a) + f′(a)(a - x)

d)

y = f(a) - f′(a)(x + a)

27.

$y=f(a)+f′(a)(x−a)$

What does this equation represent?

a)

The derivative of a function at a point.

b)

The integral of a function over an interval.

c)

The equation of a tangent line to a curve.

d)

The linear approximation of a function at a point.

28.

In an Algebra class you probably only rationalized the denominator, but you can also rationalize numerators

a)

You cannot rationalize numerators.

b)

Rationalizing numerators is not allowed in Algebra.

c)

Yes, you can rationalize numerators.

d)

Only denominators can be rationalized.

29.

For most values, the changing value for derivatives can be put at 0 to find the slope.

a)

The slope is determined by the average rate of change over an interval.

b)

The slope is always equal to the second derivative.

c)

The slope can be found by taking the integral of the function.

d)

The slope can be found by evaluating the derivative at the point of interest.

30.

You can find the derivative of a graph by just looking at it and making estimates! Isn't that fascinating!?

a)

You cannot estimate the derivative from a graph.

b)

The derivative can only be found using calculus formulas.

c)

Estimating the derivative requires numerical methods.

d)

Yes, you can estimate the derivative by looking at the graph.

31.

The sign of the function itself is completely immaterial here and will not in any way effect the sign of the derivative

a)

The sign of the function and its derivative are always the same.

b)

The sign of the function does not affect the sign of the derivative.

c)

The derivative is always positive regardless of the function's sign.

d)

The sign of the function determines the sign of the derivative.

32.

How do you find a derivative of a graph?

a)

Plot the graph on a coordinate plane.

b)

Differentiate the function representing the graph using differentiation rules.

c)

Calculate the area under the graph.

d)

Identify the maximum point of the graph.

33.

differentiate a sum or difference all we need to do is differentiate the individual terms and then put them back together with the appropriate signs. Note as well that this property is not limited to two functions

a)

The derivative of a sum or difference is the difference of the sums.

b)

The derivative of a difference is the average of the derivatives.

c)

The derivative of a sum is the product of the derivatives.

d)

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

34.

Example of “prime” notation

a)

f(x) represents the integral of the function f'(x).

b)

f''(x) represents the second derivative of the function f(x).

c)

f'(x) represents the derivative of the function f(x).

d)

f'(x) is the value of the function f(x) at a specific point.

35.

Give an example of a “fraction” notation in derivatives.

a)

Δy/Δx

b)

dy/dx

c)

∂y/∂x

d)

d/dx

36.

$(f(x)\pm g(x))′=f′(x)\pm g′(x)\ OR\ \frac{\left(d\right)}{dx}(f(x)\pm g(x))=\frac{df}{dx}\pm\frac{dg}{dx}$

What property is demonstrated here?

a)

Product rule of differentiation

b)

Quotient rule of differentiation

c)

Linearity of differentiation

d)

Chain rule of differentiation

37.

$(cf(x))′=cf′(x)\ OR\ \frac{d}{dx}(cf(x))=c\frac{df}{dx},c\ is\ any\ number$

What does this represent?

a)

The chain rule of differentiation.

b)

The linearity of differentiation.

c)

The product rule of differentiation.

d)

The quotient rule of differentiation.

38.

$If\ f(x)=c\ then\ f′(x)=0\ OR\ \frac{d}{dx}(c)=0$

What nickname would you give this rule in derivatives.

a)

The Variable Rule

b)

The Derivative Principle

c)

The Function Rule

d)

The Constant Rule

39.

$Iff(x)=x^n\ then\ f′(x)=nx^{\left(n-1\right)}\ OR\ \frac{d}{dx}(x^n)=nx^{\left(n-1\right)},n\ is\ any\ number.$

What does this represent?

a)

The sum rule of differentiation.

b)

The product rule of differentiation.

c)

The chain rule of differentiation.

d)

The power rule of differentiation.

40.

$(fg)′=f′g+fg′$

What does this represent?

a)

The sum rule for differentiation.

b)

The chain rule for differentiation.

c)

The quotient rule for differentiation.

d)

The product rule for differentiation.

41.

$(fg)′=\frac{\left(f′g−fg′\right)}{g^2}$

What does this represent?

a)

The sum rule for differentiation.

b)

The chain rule for differentiation.

c)

The product rule for differentiation.

d)

The quotient rule for differentiation.

42.

When you see radicals you should always first convert the radical to a fractional exponent and then simplify exponents as much as possible

a)

True

b)

False

43.

Whats something to know about Derivatives of Trig Functions?

a)

d/dx(tan(x)) = -sec^2(x)

b)

d/dx(cos(x)) = sin(x)

c)

d/dx(sin(x)) = -cos(x)

d)

The derivatives of trig functions include: d/dx(sin(x)) = cos(x), d/dx(cos(x)) = -sin(x), d/dx(tan(x)) = sec^2(x).

44.

$\lim_{\theta\rightarrow0}\ \frac{\left(\sinθ\right)}{\theta}$ =

a)

1

b)

0

45.

$\lim_{\theta\rightarrow0}\ \frac{\left(\cosθ-1\right)}{\theta}$ =

a)

1

b)

0

46.

$\frac{d}{dx}\sin(x)$

a)

sec^2(x)

b)

cos (x)

c)

sec(x)tan(x)

d)

-sin(x)

e)

-csc^2(x)

47.

$\frac{d}{dx}\tan\left(x\right)$

a)

sec^2(x)

b)

cos (x)

c)

sec(x)tan(x)

d)

-sin(x)

e)

-csc^2(x)

48.

$\frac{d}{dx}\cot\left(x\right)$

a)

sec^2(x)

b)

cos (x)

c)

sec(x)tan(x)

d)

-sin(x)

e)

-csc^2(x)

49.

$\frac{d}{dx}\sec\left(x\right)$

a)

-csc(x)(cot(x)

b)

sec(x)(tan(x)

50.

$\frac{d}{dx}\csc\left(x\right)$

a)

-csc(x)(cot(x)

b)

sec(x)(tan(x)

51.

$\lim_{n→∞}(1+\frac{1}{n})^n$ =

a)

1

b)

0

c)

\infty

d)

e

52.

$\lim_{h→0}\ \frac{\left(e^h−1\right)}{h}=$

a)

0

b)

e

c)

1.5

d)

1

53.

$∞\sum_{n=0}^{\infty}\frac{1}{n!}$ =

a)

1

b)

0

c)

e

d)

π

54.

For the natural exponential function, f(x)=e^x we have f′(0)= $\lim_{h\rightarrow0}\ \frac{\left(e^h−1\right)}{h}=$

a)

1

b)

1.5

c)

e

d)

0

55.

True or False: If f(x) and g(x) are inverses of each other then, than in terms of derivatives this would mean....
$g′(x)=\frac{1}{f′(g(x))}$

a)

True

b)

False

56.

$\frac{d}{dx}(e^x)=$

a)

e^x

b)

e^2x

c)

x^e

d)

e^{x+1}

57.

$\frac{d}{dx}(a^x)$

a)

a^x - ln(a)

b)

a^x * ln(a)

c)

a^x + ln(a)

d)

a^x / ln(a)

58.

$\frac{d}{dx}(\ln\left(x\right))$

(a)

59.

$\frac{d}{dx}\left(\log_ax\right)=$

a)

$\frac{1}{x\ln a}$

b)

$\frac{1}{x\ln}$

60.

$\frac{d}{dx}(\cos^{-1}x)$

a)

$\frac{1}{\left|x\right|\sqrt{x^2-1}}$

b)

$\frac{1}{\sqrt{1−x^2}}$

c)

$\frac{−1}{\left(\sqrt{1−x^2}\right)}$

d)

$\frac{1}{1+x^2}$

e)

$-\frac{1}{1+x^2}$

61.

$\frac{d}{dx}(\sin^{-1}x)$

a)

$\frac{1}{\left|x\right|\sqrt{x^2-1}}$

b)

$\frac{1}{\sqrt{1−x^2}}$

c)

$\frac{−1}{\left(\sqrt{1−x^2}\right)}$

d)

$\frac{1}{1+x^2}$

e)

$-\frac{1}{1+x^2}$

62.

$\frac{d}{dx}(\cot^{-1}x)$

a)

$\frac{1}{\left|x\right|\sqrt{x^2-1}}$

b)

$\frac{1}{\sqrt{1−x^2}}$

c)

$\frac{−1}{\left(\sqrt{1−x^2}\right)}$

d)

$\frac{1}{1+x^2}$

e)

$-\frac{1}{1+x^2}$

63.

$\frac{d}{dx}(\left(\tan\right)^{-1}x)$

a)

$\frac{1}{\left|x\right|\sqrt{x^2-1}}$

b)

$\frac{1}{\sqrt{1−x^2}}$

c)

$\frac{−1}{\left(\sqrt{1−x^2}\right)}$

d)

$\frac{1}{1+x^2}$

e)

$-\frac{1}{1+x^2}$

64.

$\frac{d}{dx}(\left(\csc\right)^{-1}x)$

a)

$\frac{-1}{\left|x\right|\sqrt{x^2-1}}$

b)

$\frac{1}{\sqrt{1−x^2}}$

c)

$\frac{−1}{\left(\sqrt{1−x^2}\right)}$

d)

$\frac{1}{1+x^2}$

e)

$-\frac{1}{1+x^2}$

65.

What are Hyperbolic Functions?

a)

Hyperbolic functions are only used in calculus.

b)

Hyperbolic functions are mathematical functions that relate to hyperbolas, analogous to trigonometric functions for circles.

c)

Hyperbolic functions are the same as logarithmic functions.

d)

Hyperbolic functions are related to ellipses, not hyperbolas.

66.

$\coth\left(x\right)$

a)

$\frac{\left(e^x−e^{-x}\right)}{2}$

b)

$\frac{\left(\sinh x\right)}{\cosh\left(x\right)}$

c)

$\frac{1}{\cos u\left(x\right)}$

d)

$\frac{\left(e^x+e^{-x}\right)}{2}$

e)

$\frac{\cosh\left(x\right)}{\sinh\left(x\right)}$

67.

$\cosh\left(x\right)$

a)

$\frac{\left(e^x−e^{-x}\right)}{2}$

b)

$\frac{\left(\sinh x\right)}{\cosh\left(x\right)}$

c)

$\frac{1}{\cos u\left(x\right)}$

d)

$\frac{\left(e^x+e^{-x}\right)}{2}$

e)

$\frac{\cosh\left(x\right)}{\sinh\left(x\right)}$

68.

$\operatorname{sech}\left(x\right)$

a)

$\frac{\left(e^x−e^{-x}\right)}{2}$

b)

$\frac{\left(\sinh x\right)}{\cosh\left(x\right)}$

c)

$\frac{1}{\cos u\left(x\right)}$

d)

$\frac{\left(e^x+e^{-x}\right)}{2}$

e)

$\frac{\cosh\left(x\right)}{\sinh\left(x\right)}$

69.

$\tanh\left(x\right)$

a)

$\frac{\left(e^x−e^{-x}\right)}{2}$

b)

$\frac{\left(\sinh x\right)}{\cosh\left(x\right)}$

c)

$\frac{1}{\cos u\left(x\right)}$

d)

$\frac{\left(e^x+e^{-x}\right)}{2}$

e)

$\frac{\cosh\left(x\right)}{\sinh\left(x\right)}$

70.

$\sinh\left(x\right)$

a)

$\frac{\left(e^x−e^{-x}\right)}{2}$

b)

$\frac{\left(\sinh x\right)}{\cosh\left(x\right)}$

c)

$\frac{1}{\cos u\left(x\right)}$

d)

$\frac{\left(e^x+e^{-x}\right)}{2}$

e)

$\frac{\cosh\left(x\right)}{\sinh\left(x\right)}$

71.

csch(x)=

a)

1/sinh(x)

b)

tanh(x)

c)

cosh(x)

d)

sech(x)

72.

$\frac{d}{dx}(\sinh x)$

a)

sec x

b)

cosh x

c)

tanh x

d)

csc x

73.

$\frac{d}{dx}(\cosh x)$

a)

cot x

b)

sec x

c)

tan x

d)

sinh x

74.

$\frac{d}{dx}(\tanh x)$

a)

sech^2(x)

b)

sinh(x)

c)

cosh(x)

d)

sec^2(x)

75.

$\frac{d}{dx}(\coth x)$

a)

-csch^2 x

b)

csch x

c)

sinh x

d)

tanh x

76.

$\frac{d}{dx}(\operatorname{sech}x)$

a)

-sech(x) * tanh(x)

b)

sech(x) * sinh(x)

c)

-sech(x) * sech(x)

d)

sech(x) * coth(x)

77.

$\frac{d}{dx}(\operatorname{csch}x)$

a)

$\operatorname{csch}x \cdot \operatorname{sech}x$

b)

$\operatorname{csch}x \cdot \operatorname{tanh}x$

c)

$-\operatorname{sech}x \cdot \operatorname{coth}x$

d)

-\operatorname{csch}x \cdot \operatorname{coth}x

78.

$F′(x)=f′(g(x))\ g′(x)$

What does this represent?

a)

The integral of a function.

b)

The limit of a function as x approaches a value.

c)

The slope of a tangent line at a point.

d)

The derivative of a composite function using the chain rule.

79.

True or False: If we have y=f(u) and u=g(x) then the derivative of y is,

$\frac{dy}{dx}=\frac{dy}{du}\left(\frac{du}{dx}\right)$

a)

True

b)

False

80.

What is a implicit differentiation

a)

Implicit differentiation is a method to differentiate equations involving multiple variables without isolating one variable.

b)

Implicit differentiation is only used for single-variable functions.

c)

Implicit differentiation requires solving for one variable first.

d)

Implicit differentiation is a technique for finding limits of functions.

81.

What is Related Rates in AP Calc BC?

a)

Related Rates focuses solely on static quantities without change.

b)

Related Rates uses only algebraic methods without derivatives.

c)

Related Rates involves finding the rate of change of one variable in relation to another using derivatives.

d)

Related Rates is only applicable to linear functions.

82.

What is a fact about differentiation of polynomials?

a)

The derivative of a polynomial is calculated by multiplying each term by its coefficient.

b)

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

c)

The derivative of a polynomial is always zero.

d)

Differentiation of polynomials requires integration techniques.

83.

$p(k)(x)=0,\ k\ge n+1$

What does this represent?

a)

A polynomial of degree k with k >= n+1 has at most n roots.

b)

A polynomial of degree k with k >= n+1 has at least n+1 roots.

c)

A polynomial of degree k with k < n has no roots.

d)

A polynomial of degree k with k = n has exactly n roots.

84.

What is higher order differentiation?

a)

Higher order differentiation is the process of taking derivatives of derivatives.

b)

Taking the first derivative of a function only.

c)

The process of finding the integral of a function.

d)

Calculating the area under a curve using limits.

85.

What is Logarithmic Differentiation about?

a)

Logarithmic differentiation is a method for differentiating complex functions using logarithms.

b)

A way to simplify polynomial expressions

c)

A method for integrating functions

d)

A technique for solving algebraic equations

86.

$\frac{d}{dx}(a^b)=0$

a)

Power Rule

b)

This is a constant

c)

Derivative of an exponential function

d)

Logarithmic Differentiation

87.

$\frac{d}{dx}(x^n)=nx^{\left(n-1\right)}$

a)

Power Rule

b)

This is a constant

c)

Derivative of an exponential function

d)

Logarithmic Differentiation

88.

$\frac{d}{dx}(a^x)=a^x\ln a$

a)

Power Rule

b)

This is a constant

c)

Derivative of an exponential function

d)

Logarithmic Differentiation

89.

$\frac{d}{dx}(x^x)=x^x\left(1+\ln x)\right)$

a)

Power Rule

b)

This is a constant

c)

Derivative of an exponential function

d)

Logarithmic Differentiation

Unit 2: Differentiation Vocab

Total questions: 89

Worksheet time: 45mins

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