Calculus Review Flashcards

A collection of flashcards aimed at reviewing calculus concepts and procedures for exam preparation.

Finding the zeros of a function

Set the function equal to zero and solve for x.

Tangent line to f(x)

Use the formula y - y₁ = m(x - x₁) with m being the derivative at point a.

Normal line to f(x)

The slope is the negative reciprocal of the tangent slope.

Increasing interval of f(x)

Where the derivative f'(x) > 0.

Increasing slope of f(x)

Where the second derivative f''(x) > 0.

Minimum value of a function

The smallest output value of f(x) on the given interval.

Minimum slope of a function

The smallest value of the derivative f'(x).

Critical values of f(x)

Points where f'(x) = 0 or is undefined.

Inflection points of f(x)

Where the second derivative changes sign.

Existence of limit

Show that $ext{lim}_{x o a} f(x)$ exists.

Continuity of f(x)

Show that $ext{lim}_{x o a} f(x) = f(a)$ and that f(a) exists.

Vertical asymptotes of f(x)

Where the denominator of a rational function equals zero but the numerator does not.

Horizontal asymptotes of f(x)

Determined by examining the limits as $x o ext{∞}$ or $x o - ext{∞}$.

Average rate of change

Given by $rac{f(b) - f(a)}{b - a}$.

Instantaneous rate of change

The derivative f'(a) at a specific point.

Average value of f(x)

Given by $rac{1}{b - a} imes ext{integral from } a ext{ to } b f(x) ext{ d}x$.

Absolute maximum of f(x)

The largest value of f(x) on the interval [a, b].

Differentiability of piecewise function

Show that the function is continuous and has no sharp points at the split.

Velocity function from position function

v(t) = s'(t), the derivative of the position function.

Finding travel distance from velocity

Integrate the velocity function on the interval [a,b].

Average velocity

The average rate of change of position over the interval [a, b].

Speeding up condition

A particle is speeding up if v(t) and a(t) have the same sign.

Rolle's Theorem conditions

f(a) = f(b) and f'(x) exists on (a, b).

Mean Value Theorem

There exists at least one c in (a, b) such that f'(c) = $rac{f(b) - f(a)}{b - a}$.

Range of f(x) on [a,b]

Set of all output values of f(x) for x in [a,b].

Range of f(x) on (-∞, ∞)

Set of all possible output values as x approaches positive and negative infinity.

Finding f'(x) by definition

Use the limit ext{lim}_{h o 0} rac{f(x + h) - f(x)}{h}.

Finding inverse derivative

g'(a) = rac{1}{f'(g(a))}.

Proportional increase of y

y is increasing in proportion to itself.

Line dividing area into two

Find the line x=c that splits area under f(x) into two equal parts.

Rate of change of population

Described by a differential equation related to population growth.

Finding area using left Riemann sums

Sum the areas of rectangles based on left endpoints.

Finding area using right Riemann sums

Sum the areas of rectangles based on right endpoints.

Midpoint rectangles

Sum the areas of rectangles based on midpoints.

Finding area using trapezoids

Use the average of heights of two endpoints to calculate the area.

Solve differential equations

Separate variables and integrate.

Meaning of integral of f(t) dt

Accumulated value of f(t) over an interval.

Cross sections perpendicular to x-axis

Perpendicular cross sections, typically square, for volume calculations.

Tangent line horizontal condition

Where f'(x)=0.

Tangent line vertical condition

Where the function is undefined.

Minimum acceleration condition

Find the minimum value of v'(t) where given v(t).

Approximation using tangent line

Use linear approximation to estimate function near a point.

Finding F(b) from F(a)

Evaluate using the Fundamental Theorem of Calculus.

Derivative of composite functions

Use chain rule $f'(g(x)) imes g'(x)$.

Taylor polynomial

A polynomial that approximates f(x) at a point.

Estimating series error

To approximate error in an alternating series, take the absolute value of the next term.

Geometric series representation

Sum representation of a geometric series.

Interval of convergence for series

Determine where the series converges.

Area between curves

Integrate to find the area bounded by f(x) and g(x).

Volume of revolution

Use the disk/washer method to calculate volume when the area between curves is rotated.

Logistic growth model

Described by the differential equation with limiting factors.

Factor polynomials using techniques

Use partial fraction decomposition.

Speed at time 0

Evaluate v(t) = 0 to find when the particle stops.

Arc length on a curve

Calculate using $L = ext{integral of } ext{sqrt}(1 + [f'(x)]^2) ext{dx}$.

Finding area inside polar curves

Use integration appropriate for polar coordinates.

Vertical tangents to a polar curve

dy/dx =0 implies vertical tangents.

Horizontal tangents to a polar curve

dy/dx = 0 implies horizontal tangents.

Euler's method approximation

Iterative method for approximating solutions to differential equations.

Convergence of a series

A series converges if the sequence of partial sums approaches a finite limit.

Polynomial approximation centered at point

Expand f(x) around a point to construct Taylor series.

Integral using substitution

Change variables to evaluate definite and indefinite integrals.

Evaluating limits involving infinity

Use L'Hôpital's Rule for indeterminate forms.

Differentiability condition of limits

Limits must be defined and finite to guarantee differentiability.

Using max/min techniques for optimization

Analyze endpoints and critical points to find maximums and minimums.

Derivative using limit definition

Use the formula f'(x) = ext{lim } rac{f(x+h) - f(x)}{h}.

Second FTC application

The second part of the Fundamental Theorem of Calculus validates the area under a curve as an antiderivative.