Differential Calculus Flashcards by MARIAH JANE SICAM

What does the derivative of a function represent?
A. The area under the curve
B. The slope of the tangent to the curve
C. The y-intercept
D. The value of the function

B. The slope of the tangent to the curve

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The derivative of a constant is:
A. 0
B. 1
C. Undefined
D. The constant itself

A. 0

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Which of the ff is a necessary condition for differentiability at a point?
A. The function must be increasing
B. The function must be continuous
C. The function must have a maximum
D. The function must be defined on an open interval

B. The function must be continuous

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If a function is differentiable at a point, then it is:
A. Always increasing
B. Continuous at that point
C. Always concave up
D. Always constant

B. Continuous at that point

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The derivative of a function at a point is the slope of:
A. The y-axis
B. The tangent line at that point
C. The secant line between two points
D. The horizontal line

B. The tangent line at that point

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The process of finding the derivative is called:
A. Integration
B. Summation
C. Differentiation
D. Approximation

C. Differentiation

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If the derivative of a function is positive on an interval, then the function is:
A. Increasing on that interval
B. Decreasing
C. Undefined
D. Concave down

A. Increasing on that interval

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The limit definition of the derivative is based on which concept?
A. Chain rule
B. Difference quotient
C. Mean value theorem
D. Power rule

B. Difference quotient

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What does it mean when the derivative of a function equals zero at a point?
A. The function is concave up
B. The function has a horizontal tangent line
C. The function is undefined
D. The function is discontinuous

B. The function has a horizontal tangent line

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Which rule is applied when differentiating a product of two functions?
A. Power rule
B. Chain rule
C. Product rule
D. Sum rule

C. Product rule

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Which of the ff is a necessary step when applying the chain rule?
A. Multiply by x
B. Differentiate both inner and outer functions and multiply
C. Take the square root of the function
D. Apply the limit

B. Differentiate both inner and outer functions and multiply

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If the graph of a function has a horizontal tangent at a point, what is true of its slope there?
A. The slope is zero
B. The slope is infinite
C. The function is undefined
D. The function has a cusp

A. The slope is zero

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The second derivative of a function gives information about:
A. Slope
B. Concavity
C. Discontinuity
D. Asymptotes

B. Concavity

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Which of the ff is an indeterminate form often seen in limits?
A. 0/0
B. 2/3
C. 1^1
D. ∞ - ∞

A. 0/0

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Which rule is most appropriate for finding the derivative of a fraction of two functions?
A. Chain rule
B. Quotient rule
C. Product rule
D. Sum rule

B. Quotient rule

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If a function has a high point (maximum) at x = a, then:
A. The function is concave up
B. The function is increasing
C. The derivative is zero at x = a
D. The second derivative is positive

C. The derivative is zero at x = a

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If the second derivative is positive on an interval, what does that say about the curve?
A. It is concave up
B. It is concave down
C. It is increasing
D. It is decreasing

A. It is concave up

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A point of inflection is a point where:
A. The slope is undefined
B. The function is constant
C. The concavity changes
D. The function has a jump

C. The concavity changes

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What does the derivative of a position function represent?
A. Acceleration
B. Distance covered
C. Velocity
D. Height

C. Velocity

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Which of the ff is not an official rule for finding derivatives?
A. Power rule
B. Secant rule
C. Chain rule
D. Product rule

B. Secant rule

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What does the second derivative of a function describe?
A. Velocity
B. Acceleration
C. Area
D. Displacement

B. Acceleration

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If the second derivative is positive over an interval, what does the graph of the function look like there?
A. It is concave up
B. It has a cusp
C. It has a vertical asymptote
D. It is linear

A. It is concave up

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What is suggested when the second derivative is equal to zero at a certain point?
A. Maximum
B. Minimum
C. Possible inflection point
D. Asymptote

C. Possible inflection point

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If a function has a local maximum at a point, what should be true about the second derivative at that point?
A. Positive
B. Negative
C. Zero
D. Undefined

B. Negative

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Which test uses the second derivative to determine whether a critical point is a maximum or minimum? A. First derivative test B. Second derivative test C. Concavity test D. Chain rule

B. Second derivative test

When the second derivative is undefined at a point, what is the most likely interpretation? A. The graph has a horizontal tangent B. The graph might have a cusp or corner C. The function has a local minimum D. The derivative is increasing

B. The graph might have a cusp or corner

What does the third derivative of a function represent in general terms? A. Position B. Velocity C. Acceleration D. Rate of change of acceleration

D. Rate of change of acceleration

In physics, if position is the original function, what does the third derivative represent? A. Velocity B. Acceleration C. Snap D. Jerk

D. Jerk

If the third derivative of a function is positive over an interval, what does this indicate about the concavity? A. The concavity is increasing B. The function is at a maximum C. The slope is zero D. The graph is linear

A. The concavity is increasing

Which of the ff must be true if the third derivative of a function equals zero at a point? A. It has a vertical tangent B. It may be a point of inflection C. The function is not continuous D. The function has a maximum there

B. It may be a point of inflection

What must be true for a function to be differentiable at a point? A. The function must be concave up B. The function must be continuous at that point C. The derivative must be zero D. The function must be constant

B. The function must be continuous at that point

Which of the ff is always true if a function is increasing? A. Its second derivative is positive B. It has a minimum C. Its first derivative is positive D. It is continuous

C. Its first derivative is positive

If a function is continuous but not differentiable at a point, what might its graph show? A. A maximum B. A cusp or sharp corner C. A vertical asymptote D. A horizontal tangent

B. A cusp or sharp corner

Which of the ff describes concave down behavior? A. The function is increasing B. The slope is constant C. The curve bends downward D. The second derivative is positive

C. The curve bends downward

If a function's slope changes from positive to negative, what can be said about the function? A. It has a local maximum B. It has a local minimum C. It is concave up D. It is undefined

A. It has a local maximum

Which term describes the value of x where the graph changes concavity? A. Derivative point B. Point of inflection C. Maximum D. Zero

B. Point of inflection

What does it mean when the graph of a function is flat (horizontal)? A. The derivative is zero B. The function is undefined C. The graph has a cusp D. The function is decreasing

A. The derivative is zero

What is the purpose of the first derivative test? A. To find inflection points B. To determine local extrema C. To find vertical asymptotes D. To determine concavity

B. To determine local extrema

Which of the ff is true about the second derivative test? A. It finds the slope of the tangent B. It determines the domain C. It identifies maxima and minima D. It tests continuity

C. It identifies maxima and minima

A function has a local minimum at a point if: A. The derivative is zero and the second derivative is positive B. The function is undefined there C. The slope is infinite D. The graph is linear

A. The derivative is zero and the second derivative is positive

If a function is continuous on an interval but has a sharp point, what can be concluded about its derivative? A. The derivative is zero B. The derivative does not exist at that point C. The second derivative is positive D. It has a maximum

B. The derivative does not exist at that point

What must be true for a point to be classified as a critical point of a function? A. The second derivative must be positive B. The function must be concave down C. The first derivative is zero or undefined D. The function must be increasing

C. The first derivative is zero or undefined

Which of the ff best describes the relationship between differentiability and continuity? A. Differentiability and continuity are unrelated B. If a function is continuous, it's always differentiable C. Differentiability implies continuity D. Continuity implies differentiability

C. Differentiability implies continuity

A function has a horizontal tangent line at a point. Which of the ff is most likely true? A. The function has a jump B. The function is not defined C. The first derivative equals zero at that point D. The second derivative is infinite

C. The first derivative equals zero at that point

Which of the ff guarantees at least one point where the derivative is zero between two points where the function has equal values? A. Chain Rule B. Rolle’s Theorem C. Product Rule D. Extreme Value Theorem

B. Rolle’s Theorem

A function is increasing and concave down on an interval. What does this say about its derivatives? A. Both are positive B. Both are negative C. First derivative positive, second derivative negative D. First derivative negative, second derivative positive

C. First derivative positive, second derivative negative

If a function’s graph levels off as x approaches infinity, what can be inferred about its derivative? A. The derivative is increasing B. The derivative approaches zero C. The derivative is undefined D. The function is non-differentiable

B. The derivative approaches zero

What can be said about the second derivative if a function has a local maximum and is smooth? A. It is undefined B. It is negative C. It is positive D. It is zero

B. It is negative

If a function's derivative is increasing, what is true about the function? A. The function is concave up B. The function is constant C. The function has no inflection points D. The function is decreasing

A. The function is concave up

A function has a point where its slope changes rapidly, but the curve remains smooth. What might be true at that point? A. The function has a maximum B. The derivative does not exist C. The second derivative is large in magnitude D. The function is not continuous

C. The second derivative is large in magnitude

When solving related rates problems, which of the ff is typically the first step? A. Differentiate with respect to time B. Identify known and unknown variables C. Guess the formula D. Plug in the numbers

B. Identify known and unknown variables

What does it mean if a quantity’s rate of change is negative? A. The quantity is increasing B. The quantity is decreasing C. The quantity is constant D. The quantity is doubling

B. The quantity is decreasing

In a related rates problem, why must we apply the chain rule when differentiating with respect to time? A. Because time is not part of the problem B. Because all variables are functions of time C. Because it’s easier D. Because the second derivative is needed

B. Because all variables are functions of time

A balloon is rising vertically, and its shadow moves horizontally due to the sun. What type of relationship exists between the balloon and its shadow? A. No relationship B. Related rates C. Time-independent D. Circular motion

B. Related rates

A ladder slides down a wall. Which quantity decreases over time? A. The height of the wall B. The length of the ladder C. The height of the ladder’s top from the ground D. The thickness of the ladder

C. The height of the ladder’s top from the ground

In related rates, if volume increases while height stays the same, what is likely changing? A. Nothing B. The radius or width C. The slope D. The area under the curve

B. The radius or width

When two objects are moving in perpendicular directions, which theorem is typically used to relate them? A. Chain Rule B. Pythagorean Theorem C. Mean Value Theorem D. Fundamental Theorem of Calculus

B. Pythagorean Theorem

A camera tracks a car moving along a straight road. As the car moves, what rate is changing for the camera? A. The car’s speed B. The length of the road C. The angle of the camera D. The size of the car

C. The angle of the camera

In a related rates problem involving a cone, which variable typically changes over time? A. Volume only B. Radius only C. Volume, radius, and height D. Temperature and radius

C. Volume, radius, and height

Why must you wait until after differentiating to substitute numerical values in a related rates problem? A. To simplify the math B. It’s easier to guess the answer C. To apply the chain rule correctly D. To graph the function

C. To apply the chain rule correctly

Which of the ff best explains why a function could be continuous at a point but not differentiable? A. The slope is zero B. There’s a sharp corner or cusp C. It is linear D. It has a vertical tangent

B. There’s a sharp corner or cusp

Which of the ff best describes a function that is differentiable everywhere? A. Smooth and continuous with no sharp points B. It is always increasing C. It has vertical tangents D. It is constant

A. Smooth and continuous with no sharp points

What does a cusp on a function’s graph indicate about its behavior? A. The slope is zero B. It has a maximum C. It is not differentiable at that point D. The function is concave up

C. It is not differentiable at that point

If a function has a vertical tangent at a point, what can be said about its derivative there? A. The derivative is undefined B. The derivative is zero C. The function has a maximum D. The second derivative is negative

A. The derivative is undefined

Which of the ff characteristics implies differentiability at a point? A. The graph is smooth at the point B. The function is increasing C. It has a cusp D. The graph has a jump

A. The graph is smooth at the point

What kind of function is always differentiable over its domain? A. Step function B. Polynomial function C. Absolute value function D. Piecewise function with jumps

B. Polynomial function

What happens to the derivative of a function at a point where the graph has a vertical tangent? A. The slope is zero B. The function is increasing C. The derivative is undefined D. The second derivative is negative

C. The derivative is undefined

If a function has different left-hand and right-hand slopes at a point but is continuous there, what conclusion can be made? A. The limit does not exist B. The function is differentiable C. The derivative does not exist at that point D. The function is constant

C. The derivative does not exist at that point

If the left-hand and right-hand derivatives are unequal at a point, what does this imply? A. The function is not differentiable at that point B. The function is not continuous C. The second derivative is negative D. The function has a maximum

A. The function is not differentiable at that point

What does the function being differentiable mean, in practical terms? A. It must have vertical tangents B. It is smooth and has no sharp corners C. It is always increasing D. It has a hole

B. It is smooth and has no sharp corners

What does it mean when the limit of a function as x approaches a value exists? A. The derivative exists there B. The function approaches the same value from both sides C. The function is undefined D. The graph has a vertical asymptote

B. The function approaches the same value from both sides

Which of the ff situations means a limit does not exist at a point? A. The function approaches different values from left and right B. The graph is smooth at that point C. The derivative is zero D. The graph has a hole

A. The function approaches different values from left and right

What is true if the graph of a function approaches the same height from both sides but has a hole at that point? A. The limit exists but the function is undefined there B. The derivative is undefined C. The function is discontinuous D. The graph is vertical

A. The limit exists but the function is undefined there

A function has a jump discontinuity at x = 2. What can be said about the limit at x = 2? A. The limit does not exist B. It equals the function value C. It is infinite D. The graph is concave

A. The limit does not exist

The limit of f(x) as x approaches 5 is 12. What does this mean graphically? A. The function stops at x = 5 B. The function is increasing C. The y-value of the graph approaches 12 as x → 5 D. The function has a hole

C. The y-value of the graph approaches 12 as x → 5

What must be true about a function in order for it to be continuous at a point x = a? A. The function must be linear B. The limit must exist and equal the function value C. The derivative must be zero D. The graph must be concave

B. The limit must exist and equal the function value

Which of the ff causes a limit to be infinite at a point? A. The function is linear B. The graph has a vertical asymptote C. The function has a maximum D. The function is constant

B. The graph has a vertical asymptote

What does a removable discontinuity look like on a graph? A. A sharp turn B. A hole in the graph C. A vertical asymptote D. A curved section

B. A hole in the graph

What is the best reason for using limits in calculus? A. To avoid fractions B. To understand behavior near specific points C. To integrate functions D. To solve equations

B. To understand behavior near specific points

If the limit of a function as x approaches a number exists, but the function value at that number is different, what is true? A. The function has a removable discontinuity B. The function is differentiable C. The limit does not exist D. The function is linear

A. The function has a removable discontinuity

A function f(x) is defined everywhere, and the limit of f(x) as x approaches 3 exists. Which of the ff statements is true? A. f(3) must equal the limit B. The value of f(3) is not needed for the limit to exist C. f(3) does not exist D. The function is undefined at x = 3

B. The value of f(3) is not needed for the limit to exist

Which of the following best defines the derivative of a function? A.) The slope of the tangent line at a point B.) The area under the curve C.) The maximum value of a function D.) The average rate of change

A.) The slope of the tangent line at a point

Why is the derivative of a constant function zero? A.) Because the graph is a straight horizontal line B.) Because it is a polynomial function C.) Because its slope is always positive D.) Because it has no domain

A.) Because the graph is a straight horizontal line

What does the second derivative of a function represent? A.) The slope of the function itself B.) The curvature or concavity of the graph C.) The area under the curve D.) The inverse of the first derivative

B.) The curvature or concavity of the graph

Differential Calculus Flashcards

(85 cards)