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AP Calculus Formulas
For AB only
| Question | Answer |
|---|---|
| What is tan(x) = ? | sin(x) / cos(x) opposite / adjacent 1 / cot(x) |
| What is cot(x) = ? | cos(x) / sin(x) adjacent / opposite 1 / tan(x) |
| What is sin(pi) = ? | 0 |
| What is cos(pi) = ? | -1 |
| What is sin^2(x) + cos^2(x) = ? | 1 |
| What is the derivative of sin(x)? | cos(x) |
| What is the derivative of cos(x)? | -sin(x) |
| What is the derivative of e^x? | e^x |
| What is the derivative of cot(x)? | -csc^2(x) |
| What is the derivative of a^x? | (a^x)(lna) |
| What is the derivative of a^g(x)? | (a^g(x)) (ln(a)) (g'(x)) |
| What is the derivative of tan(x)? | sec^2(x) |
| What is the derivative of sin^-1(x)? | 1 / sqrt(1 - x^2) |
| What is the derivative of cos^-1(x)? | -1 / sqrt(1 - x^2) |
| What is the derivative of sec(x)? | sec(x)tan(x) |
| What is the derivative of tan^-1(x)? | 1 / (1 + x^2) |
| What is the derivative of csc(x)? | -csc(x)cot(x) |
| What is the derivative of ln(x)? | 1 / x |
| What is the derivative of f^-1(x) | 1 / f' (f^-1(x)) |
| What are the steps to solving for a maximum or minimum x-value given a position function, f(x)? | - take the derivative - determine the critical points by setting f'(x) = 0 - determine the signs of the intervals before and after the critical points - positive to negative means max, negative to positive means min |
| How can we determine when a position function, f(x), is concave up or concave down? | When its acceleration function, f''(x) is positive for concave up or negative for concave down |
| What is an inflection point and how do we determine one? | A point where concavity changes sign. Check the intervals surrounding the critical points from the second derivative (aka possible points of inflection) to see how the signs change. |
| What type of function do you need in order to apply the Mean Value Theorem? | A differentiable function (which is automatically continuous). |
| What type of function do you need in order to apply the Intermediate Value Theorem? | A continuous function. |
| What does it mean when a function is continuous? | The function has defined points over the given interval and all limits exist. |
| What does it mean when a point is defined? | When an x-value has a real y-value. |
| When you're asked to approximate or estimate a value, you should use the... | slope formula |
| When asked to solve for the average value, you should use | 1 / (b - a) integral (a to b) of f(x) dx |
| What asked to solve for the average rate of change, you should use the... | slope formula |
| When asked for the rate of change for f(x), you're solving for | f'(x) |
| When asked for the rate of change of the rate of f(x), you're solving for | f''(x) |
| Integrating the acceleration function gives you the... | velocity function |
| When asked to determine if the speed of a particle is increasing or decreasing at a given time, you must check to see | if the velocity and acceleration function have the same sign or different signs at that given time |
| When you're asked to find the total distance, which function are you integrating? | You are integrating the absolute value of a velocity function |
| If you want to determine if an object changed directions, you focus on which derivative? | the first derivative (velocity) |
| To write the equation of a tangent line, we can use what formula? | y - y_1 = m (x - x_1) |
| How do we find the area between two functions? | integral (a to b) of top function - bottom function dx |
| How can we show a piecewise function is continuous at a given x-value? | - check to see if the point is defined - check to see if the left and right handed limit are defined and equal - check to see if all three of the above are equal |
| Whenever you are looking for an absolute maximum or minimum value, you must test which points? | the critical points and the endpoints, if given |
Created by:
frahman
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