AB Calculus Study Guide: Limits, Derivatives, and Integrals

What is a limit in calculus?

The value that a function approaches as the variable within the function gets closer to a particular value.

How can you find the limit of a simple polynomial?

By plugging in the number that the variable is approaching.

What does it mean if a graph approaches two different values for the same number?

The limit does not exist.

What is one method to estimate limits?

Using a table of values.

What is the Squeeze Theorem?

If g(x) ≤ f(x) ≤ h(x) for all x in an interval containing a, and lim g(x) = L and lim h(x) = L, then lim f(x) = L.

What is the limit of sin(x)/x as x approaches 0?

1

What is a jump discontinuity?

Occurs when the curve breaks at a point and starts somewhere else, with limits from the left and right existing but not matching.

What characterizes an essential or infinite discontinuity?

The curve has a vertical asymptote.

What is a removable discontinuity?

An otherwise continuous curve has a hole in it that can be filled to remove the discontinuity.

What are the conditions for continuity at x=c?

f(c) exists, the limit as x approaches c exists, and lim f(x) = f(c).

What is a vertical asymptote?

A line that a function cannot cross because the function is undefined there.

What is a horizontal asymptote?

The end behavior of a function, which can be crossed.

What happens if the highest power of x is in the numerator of a rational expression?

The limit as x approaches infinity is infinity, indicating no horizontal asymptote.

What happens if the highest power of x is in the denominator of a rational expression?

The limit as x approaches infinity is zero, and the horizontal asymptote is the line y=0.

What is the Intermediate Value Theorem (IVT)?

If a function f(x) is continuous on the interval [a,b] and C is between f(a) and f(b), then there is at least one number in [a,b] such that f(x) = C.

What is the difference quotient?

The formula y2 - y1 / x2 - x1 used to find the average rate of change.

What is the instantaneous rate of change?

The limit of the difference quotient as h approaches 0.

How do you approximate the slope of a curved line?

By using the secant line and the difference quotient.

What is the definition of a derivative?

The limit of the difference quotient as the interval approaches zero.

What is the relationship between limits and asymptotes?

Limits help determine the behavior of functions near vertical and horizontal asymptotes.

What is the significance of removable discontinuities in functions?

They can be eliminated by redefining the function to fill the hole.

What is the importance of continuity in calculus?

A function must be continuous at every point in an interval to be considered continuous on that interval.

What is the definition of the derivative?

The derivative is the rate of change at a specific point, found using the limit as h approaches 0 in the difference quotient.

What is the notation for the first derivative of a function f(x)?

f'(x)

What is the notation for the second derivative of a function f(x)?

f''(x)

What is the Constant Rule in differentiation?

If f(x) = k (where k is a constant), then f'(x) = 0.

What does the Constant Multiple Rule state?

If you have a constant multiplied by a function, you can pull the constant out: [k * f(x)]' = k * f'(x).

What is the Power Rule for differentiation?

If f(x) = x^n, then f'(x) = nx^(n-1).

How can the Power Rule be described?

Multiply down the exponent and decrease the power.

What is the Product Rule for differentiation?

If f(x) = uv, then f'(x) = (u)(dv/dx) + (v)(du/dx).

What is the mnemonic for the Product Rule?

1d2 + 2d1 (first times derivative of second plus second times derivative of first).

What is the Quotient Rule for differentiation?

If f(x) = u/v, then f'(x) = (v)(du/dx) - (u)(dv/dx) / v^2.

What is the mnemonic for the Quotient Rule?

Low d high - high d low / low^2.

What are 'memory derivatives'?

These are derivatives that are easier to memorize than to derive, such as those of sin(x), cos(x), e^x, and ln(x).

What is the Chain Rule in differentiation?

If y = f(g(x)), then y' = f'(g(x)) * g'(x).

What is an alternative form of the Chain Rule?

If y = y(v) and v = v(x), then dy/dx = (dy/dv) * (dv/dx).

What is the personal memory trick for the Chain Rule?

'Douter, inner, dinner' - drop the power down to outside the parentheses, leave the inner, multiply by the derivative of the inner.

What is implicit differentiation?

It is used when you can't isolate y in terms of x, allowing you to find the derivative in terms of both variables.

How is implicit differentiation expressed?

(dx/dy) = (1/(dy/dx)).

What is the process for implicit differentiation using the example x^2 + y^2 = 25?

Differentiate both sides with respect to x: 2x + 2y(dy/dx) = 0, then solve for dy/dx.

What is the slope of the tangent line at the point (3, 4) for the equation x^2 + y^2 = 25?

The slope dy/dx = -3/4.

What does the tangent line represent in calculus?

The tangent line touches the curve at exactly one point and represents the instantaneous rate of change.

What is the significance of the limit in the definition of the derivative?

It allows us to find the slope as x approaches an infinitesimally small value.

What happens to the accuracy of the slope as points get closer together?

The slope becomes more accurate.

What is the role of the Instantaneous Rate of Change in finding derivatives?

It is used to derive the tangent line that represents the slope at a specific point.

What is the slope of the tangent line at the point (3, 4)?

The slope of the tangent line is -3/4.

What is the equation of the tangent line at the point (3, 4) with a slope of -3/4?

y - 4 = -3/4(x - 3).

How do you find the derivative of an inverse function at a point?

Take the reciprocal of the derivative at the corresponding y value.

If you want to find g'(1) for the inverse function at (1, 2), what steps do you take?

Find f'(2) and take the reciprocal.

What is the relationship between f'(x) and slope?

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

What should you use when two terms are multiplied together in differentiation?

Use the product rule unless it's easier to multiply it out.

What rule should you apply if there is a function within another function?

You will almost certainly need to use the chain rule.

What is the derivative of position with respect to time?

Velocity, measured in meters per second (m/s).

What is the derivative of velocity with respect to time?

Acceleration, measured in meters per second squared (m/s²).

How can you find acceleration from a velocity function v(t)?

Take the derivative of the velocity function with respect to time.

What is the acceleration of the particle at time t=2 if v(t) = 3t² - 4t + 2?

The acceleration is 8 m/s².

What does it mean when the signs of velocity and acceleration match?

Particles will speed up when both velocity and acceleration are either negative or positive.

What does the derivative tell us in the context of motion?

It tells us the change of a unit over time, such as position or velocity.

What is the formula for the volume of water in a pool as a function of time?

V(t) = 8t² - 32t + 4, where V is in gallons and t is in hours.

How do you find the rate of change of the volume of water in the pool?

Take the derivative dV/dt = 16t - 32 gallons per hour.

At what time does the volume of water in the pool stop changing?

At t=2 hours, the volume isn't changing.

What is the temperature function of a cup of coffee after t minutes?

x(t) = 70 + 50e^(-0.1t).

How do you find the rate of change of temperature with respect to time at t=5 minutes?

Take the derivative of the temperature function x(t) with respect to time.

What does the derivative of a function represent?

The rate of change of that function.

What is the relationship between position, velocity, and acceleration in motion?

Position is the integral of velocity, and velocity is the integral of acceleration.

What is the significance of the second derivative in motion?

It represents acceleration, indicating how velocity changes over time.

What is the derivative of x(t) = -5e^(-0.1t)?

d/dt of x(t) = -5e^(-0.1t)

What is the value of the derivative x'(5)?

x'(5) = -5e^(-0.1(5)) ≈ -2.27

What is the relationship between related rates in calculus?

Related rates problems involve finding the rate of change of one quantity in relation to another.

How do you find the rate of change of the radius of a pool expanding at 16π square inches per second when the radius is 4 inches?

Using the formula A = πr^2, we differentiate to get dA/dt = 2πr(dr/dt) and solve 16π = 2π(4)dr/dt to find dr/dt = 2 inches per second.

What is the formula for the volume of a sphere?

V = (4/3)πr^3.

How do you find the rate of change of the radius of a balloon being inflated at 10 cubic inches per second when the radius is 4 inches?

Using dV/dt = 4πr^2(dr/dt), substitute dV/dt = 10 and r = 4 to find dr/dt = 10/(16π) inches per second.

What are the steps to solve related rates problems in calculus?

1. Read the problem carefully and identify all given information. 2. Draw a diagram if possible. 3. Determine what needs to be found and assign a variable to it. 4. Write an equation that relates the variables involved. 5. Differentiate both sides with respect to time. 6. Substitute in the given values and solve for the unknown rate. 7. Include units in the final answer and check that it makes sense.

What is a differential in calculus?

Differentials are very small quantities that correspond to a change in a number, denoted by Δx.

How can differentials approximate the value of a function?

Using the approximation f(x + Δx) ≈ f(x) + f'(x)Δx.

How do you approximate (3.98)^4 using differentials?

Let f(x) = x^4, x = 4, Δx = -0.02, then use f(x + Δx) ≈ f(x) + f'(x)Δx to find the approximation.

What is L'Hospital's Rule?

L'Hospital's Rule states that if a limit gives 0/0 or ∞/∞, you can take the derivative of the numerator and denominator and try again.

What happens when you apply L'Hospital's Rule to the limit of (5x^3 - 4x^2 + 1)/(7x^3 + 2x - 6) as x approaches infinity?

After applying L'Hospital's Rule twice, the limit simplifies to 30/42 or 5/7.

What does the Mean Value Theorem (MVT) state?

The MVT links the average rate of change and the instantaneous rate of change, asserting there is a point where the slope of the tangent equals the slope of the secant line.

What is Rolle's Theorem?

Rolle's Theorem is a special case of the MVT, stating that a continuous, differentiable curve has a horizontal tangent between any two points.

What does the Extreme Value Theorem state?

If a function is continuous on a closed interval, it must have both a maximum and a minimum value on that interval.

What are the two types of extrema?

Absolute (global) and Local (relative)

What defines an absolute extrema?

No other value is higher/lower than an absolute.

What defines a local extrema?

The highest/lowest in the nearby area.

What are critical points?

Any place an extrema could exist.

How can you determine intervals of increase and decrease in a function?

Using the first derivative.

What does it mean if f'(x) > 0?

The function is increasing.

What does it mean if f'(x) < 0?

The function is decreasing.

How do you find critical numbers?

Take the derivative of the function and set it equal to zero.

What is the first step in analyzing the function f(x) = x^3 - 6x^2 + 9x + 2 for increasing/decreasing intervals?

Take the derivative, f'(x) = 3x^2 - 12x + 9.

What are the critical points for the function f(x) = x^3 - 6x^2 + 9x + 2?

x = 1 and x = 3.

What does it indicate if the first derivative shifts from positive to negative?

There will be a relative maximum.

What does it indicate if the first derivative shifts from negative to positive?

There will be a relative minimum.

How do you find absolute extrema?

Consider the endpoints and critical numbers, then evaluate the original function.

What does the second derivative tell us about a function?

It indicates whether the function is increasing or decreasing at an increasing or decreasing rate.

What does f''(x) > 0 indicate about the function?

The function is concave up.

What does f''(x) < 0 indicate about the function?

The function is concave down.

How do you find points of inflection?

Set the second derivative equal to zero and find where it changes sign.

For the function f(x) = x^3 - 6x^2 + 9x + 2, what is the second derivative?

f''(x) = 6x - 12.

What is the critical point for concavity of the function f(x) = x^3 - 6x^2 + 9x + 2?

x = 2.