AP BC Calc: Khan Academy Review

What is the integral of csc²(x)?

-cot(x) + C

What is the integral of csc(x)cot(x)?

-csc(x) + C

What is the derivative of cscx?

-cscxcotx; x≠πn, n∈Z

What is the derivative of cotx?

-csc²x; x≠πn, n∈Z

What is the integral of e^-x

-e^-x

What is the derivative of cosx?

-sinx; -∞<x<∞

What is the volume of a sphere?

V = (4/3)πr³

What is the volume of a cylinder?

V=πr²h

What is the derivative of position?

Velocity.

Volume of a cube.

Volume = length x width x height. s³

What is a point of inflection?

When a function switch from concave up to concave down or vise versa.

How do you find points of inflection?

When f''(x) switches signs, or when f''(x) = 0 or is undefined.

The function C gives the costs (in dollars) of producing x liters of a certain sauce. What is the best interpretation for the following statement? The slope of the line tangent to the graph of C at x=12 is equal to 4.

When the amount of sauce produced is 12 liters, the costs increase at a rate of 4 dollars per liter.

Where could we determine inflection points on a graph of f''(x)?

Where it crosses the x-axis would show inflection points. It has to cross, not just touch.

What is the derivative of sinx?

cosx; -∞<x<∞

In regards to related rates, what is dA/dt?

dA/dt = π2r*dr/dt

What is the derivative of e^x?

e^x

If f''(c) is positive (>0), what do we know about f?

f has a relative minimum at x=c. Same logic can be applied for f''(x) < 0.

How can we determine if a function is concave down/up?

f''(x) = +, then concave up f''(x0 = -, the concave down

How do we know when a function is decreasing?

f'(x) < 0

What are the 3 strategies for calculating limits?

f(a) = b/0 --> asymptote (probably) f(a) = b --> limit found (probably) f(a) = 0/0 --> indeterminate form --> factoring, conjugates, or trig identities --> approximation

What is an appropriate calculus-based justification (using g') for the fact that g has a relative minimum point at x = c?

g' crosses the x-axis from below it to above it at x = c.

If g(x) = f^(-1)(x), what is g(f(x))?

g(f(x)) = x

What is a justification for h having a relative maximum at x=-4?

h''(-4) is negative.

If h is the inverse of f, what is h'(x)?

h'(x) = 1/(f'(h(x))

What is the integral of 1/(a²+x²)?

(1/a)arctan(x/a) + C

What is ln(1)?

0

What is the antiderivatives of sinx and cosx?

-cosx and sinx (+ C)

What is the derivative of logx?

1/((ln10)x) *10 is the base

What is the derivative of log∨a(x)?

1/((lna)x)

What is the derivative of ln(x)?

1/x; (0,∞)

What are critical points?

A critical point is a point in the domain of the function where the function is either not differentiable or the derivative is equal to zero.

How can we tell if an object is speeding up, slowing down, or neither on a graph of velocity.

Above the x-axis: If the slope is positive, the speed is increasing; if the slope is negative, the speed is decreasing; if slope is 0, neither. If the point lies on the x-axis of a velocity graph, it is neither speeding up nor slowing down. Below the x-axis: If the slope is negative, speed is increasing; if slope is positive, speed is decreasing; if slope is 0, neither.

What is the derivative of velocity?

Acceleration.

What is the best interpretation or the following statement? A tank is being drained of water. The function V gives the volume of liquid in the tank, in liters, after t minutes. The slope of the line tangent to the graph of V at t = 7 is equal to -3.

After 7 minutes, the tank is being drained at an instantaneous rate of 3 liters per minute.

When the function f is decreasing, the antiderivative is...

Concave down.

When the function f is increasing, the antiderivative is...

Concave up.

When the function f is negative, the antiderivative is...

Decreasing.

Eden walked at a rate of r(t) kilometers per hour (where t is the time in hours). What does the integral from 2 to 3 r(t)dt = 6 mean?

Eden walked 6 kilometers during the third hour.

When the function f changes sign/crosses the x-axis, the antiderivative is...

Extremum point.

What do you have to do to optimize?

Find the critical points of a function and determine if they are minimum or maximum points (usually by determining the concavity with the second derivative).

What do the existence theorems guarantee (how are they different)?

IVT guarantees a point where the function has a certain value between two given values. EVT guarantees a point where the function obtains a maximum or a minimum value. MVT guarantees a point where the derivative has a certain value.

What is the best interpretation or the following statement? Eddie drove from New York City to Philadelphia. The function D gives the total distance Eddie has driven (in kilometers) t hours after he left.

If D is the distance driven, D' is the instantaneous rate of change.

Intermediate Value Theorem

If f is continuous on [a,b] and k is a number between f(a) and f(b), then there exists at least one number c such that f(c)=k

How can we justify that a function is increasing?

If its derivative is positive, the function is increasing.

What do we know about position at the point on a velocity graph where the slope is 0?

If the slope is zero on a velocity graph, position is changing concavity.

If acceleration is positive, what do we know about the particle's movement/speed?

If the velocity and acceleration are different signs, speed is decreasing. If they are the same signs, speed is increasing.

What do we know about position if the velocity curve is below the x-axis?

If velocity is below the x-axis, the position is decreasing (moving backward). If velocity is above the x-axis, position in increasing (forward). If velocity is on the x-axis, position is stopped (neither forward nor backward).

When the function f is positive, the antiderivative is...

Increasing.

When the function f is an extremum point, the antiderivative is...

Inflection point.

How does the fundamental theorem of calculus apply to a integral from x to b (not a to x)—when x is on the bottom rather than the top.

It applies the same way, but the answer is negative.

If velocity is negative, which direction is the particle moving?

Left.

How can we tell that a point is a max by derivatives?

Max if it goes from f'(x) > 0 to f'(x) < 0

How can we determine absolute/global extrema?

Plug in the critical points and end points into the original function (not the derivative) and compare which goes highest/lowest.

Surface area of a cube.

S = 2x² + 4xh

When given a logistic differential equation, how can we determine when it is increasing the fastest?

Set the rate equal to zero, and then find the average of the two values. I.e. The maximum of dP/dt is obtained when the population is equal to the P-value of the vertex of the parabola. We can find the P-value of the vertex by finding the roots and taking their average.

How do you find critical points?

Take the derivative and set it equal to zero (or figure out where the derivative is undefined).

What is the extreme value theorem?

The extreme value theorem states that if a real-valued function f is continuous on the closed interval [a, b], then f must attain a maximum and a minimum, each at least once.

What is the intermediate value theorem?

The intermediate value theorem states that if f is a continuous function whose domain contains the interval [a, b], then it takes on any given value between f(a) and f(b) at some point within the interval.

What must be true for a function to be continuous at a point?

The limit and the exist must exist and be equal.

What is the mean value theorem?

The mean value theorem states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints.

How does a = e^(lna)

The natural log of a is the power you need to raise e to to get to a.

What are the conditions under the MVT?

The precise conditions under which MVT applies are that f is differentiable over the open interval (a,b) and continuous over the closed interval [a,b]. Since differentiability implies continuity, we can also describe the condition as being differentiable over (a,b) and continuous at x=a and x=b. For MVT to apply, the function must be differentiable over the relevant interval, and continuous at the interval's edges.

What is the integral of 1/(√(a²-x²))?

arcsin(x/a) + C

What is the derivative of a^x?

ln(a)a^x

What is the indefinite integral of 1/x?

ln(x)

lnx=?

log∨e(x)

What is the derivative of xⁿ?

nxⁿ⁻¹

What is the integral of sec(x)tan(x)?

sec(x) + C

What is the derivative of secx?

secxtanx; x≠π/2+πn, n∈Z

What is the derivative of tanx?

sec²x; x≠π/2+πn, n∈Z

What is the integral of sec²(x)?

tan(x) + C

If y = cos^(-1)x, what is x?

x = cosy