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AP Calculus Review

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Section 1

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d/dx secx

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Calculus Math

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Mar 1, 2020

Cards (38)

Section 1

(38 cards)

d/dx secx

Front

secxtanx

Back

d/dx sin(x)

Front

= cosx

Back

Linearization (linear approximation)

Front

fancy way to say "plug an x-value into a tangent line"

Back

Chain Rule

Front

h(x)=f(g(x)) h'(x)=f'(g(x))*g'(x)

Back

d/dx ln(x)

Front

1/x

Back

Differentiability

Front

Whether or not we can differentiate at a particular x-value. We can NOT differentiate where there are corners, cusps, discontinuities, or vertical tangent lines.

Back

Concavity

Front

The shape of the graph. Find the second derivative at the point of interest. it it's positive, the function is concave up there. If it's 0, you may have what's called an inflection point.

Back

Intermediate Value Theorem

Front

States that if a function is continuous on a closed interval, then the function attains all the y-values between the endpoints.

Back

Candidates Test

Front

Lets us find the maximum or minimum on a closed interval. 1. Find your candidates (CP and endpoints) 2. Plug each candidates back into the original equation 3. Pick the biggest/smallest This finds the absolute extreme

Back

Infinite Limits

Front

Limits whose answer is infinity or -infinity. These are vertical asymptotes. Direct substitution yields #/0. Then think about each sided-limit to determine if it's negative or positive. These limits are how you justify vertical asymptotes.

Back

Limits at Infinity

Front

The behavior of a function for large values of x; potentially horizontal asymptotes if the limit is finite. Use dominance: exponentials beat polynomials, polynomials beat logarithms. If there's a tie, use the ratio of the coefficients. (Watch out for negative exponents at infinity)

Back

Second Derivative Test

Front

Another way to determine whether or not we have a local max or min. 1) Find CP 2) Plug these points into the second derivative 3) State whether it's a max or a min. If the second derivative is positive, that means concave up. That means it's a min. If the second derivative is negative, that means concave down. That means a max

Back

Derivatives

Front

The instantaneous rate of change for a function/slope of tangent line. Can be approximated using average rate of change on a small interval (f(b)-f(a)/b-a).

Back

Limits

Front

The y-value a function approaches at a certain x-value.

Back

d/dx cot(x)=

Front

-csc^2x

Back

Critical Point

Front

Places on the function where the slope is 0. (Find where the derivative is 0; make sure the 0s are in the domain)

Back

Quotient Rule

Front

h'(x)= (g(x)f'(x)-f(x)g'(x)/g(x)^2)

Back

Indeterminate Form

Front

Limits in indeterminate form are either 0/0 or infinity/infinity when trying direct substitution. Solve these with L'Hopital's Rule.

Back

d/dx loga(x)

Front

1/ln(a) * 1/x

Back

Tangent line

Front

A line that shares a point and a slope with a function. y-y1=m(x-x1) y1=f(x1) m=f'(x1)

Back

Product rule

Front

h'(x) = f'(x) g(x) + g'(x) f(x)

Back

Mean Value Theorem

Front

Lets us determine the existence of a derivative value. If the function is continuous and differentiable, then there exists a value of x such that f'(c)=f(b)-f(a)/b-a

Back

One-sided limits

Front

A limit where you only approach the x-value from one side. Usually direct substitution will still work. Sometimes we have to be careful with signs so we think about what y-values look like just before or after.

Back

Power Rule

Front

The derivative formula for basic polynomials. f(x)=x^n, f'(x)=nx^n-1. This can be done with a long polynomial piece-by-piece. If you have a coefficient, just multiply the coefficient by the exponent. The derivative of a constant is always 0.

Back

Graphs of Derivatives

Front

Whenever the function is increasing, the derivative is positive. Whenever the function is decreasing, the derivative is negative. Whenever the function is concave up, the derivative is increasing. Whenever the function is concave down, the derivative is decreasing.

Back

d/dx csc(x)

Front

-cscxcotx

Back

Optimization

Front

Lets us find the largest or smallest of something we're interested in. Word problems asking for max, min, most, least

Back

Inflection Point

Front

Where the graph changes concavity. Find were the second derivative changes from + to - or vice versa. Do this by setting the second derivative equal to 0 and then making a sign chart. It is only an IP if the second derivative changes sign here

Back

Related Rates

Front

When two variables are changing with respect to a third, we can differentiate implicitly to find particular rates of change. Differentiate with respect to the variable of interest. Then plug in what you know, solve for what you don't.

Back

Discontinuity

Front

Breaks in the graph. We prove discontinuity be showing a function breaks one: limit exists from both sides, f(a) exists, lim f(x)=f(a). Types: infinite (VA), removable (holes), jump (piecewise).

Back

Implicit Differentiation

Front

Differentiating with respect to a different variable. (Use chain rule)

Back

d/dx a^x d/dx e^x

Front

f'(x)= ln(a)a^x f'(x)=e^x

Back

Particle Motion

Front

Usually asked about a particle moving along the x-axis. x(t) is position, v(t) velocity, a(t) is acceleration. Displacement x(b)-x(a), average velocity using average rate of change x(b)-x(a)/b-a. If velocity is negative, we're moving to the left. If velocity is positive, moving to the right. If acceleration and velocity match sign, speeding up. If acceleration and velocity mismatch, slowing down. Greatest speed is the largest velocity, positive or negative.

Back

d/dx cos(x)

Front

-sin(x)

Back

Direct Substitution

Front

A limit where the x-value is in the domain of the function. (Plug the number straight in)

Back

d/dx tanx

Front

sec^2x

Back

First Derivative Test

Front

Lets us determine whether or not there's a relative max or min. 1) Find critical points 2) Make a sign chart 3) State whether or not it's a max or min. This sign chart is NOT your explanation. You must explain that the point "is a max because the derivative goes from + to - here" or "is a minimum because the derivative goes from - to + here"

Back

Continuity

Front

Graphically, it basically means you can trace the function without picking up your pencil. Mathematically, it means the limit at that point is the same as the function's value. (limit exists from both sides, f(a) exists, limit=f(a).

Back