AP Calculus Midterm

Derivative Rules, Theorum Definitions, and derivative manifestations

When f’(x)>0, f is….

increasing

When f’(x)<0, f is….

decreasing

When f’(x)=0, f has a…

local maximum or minimum

When f’’(x)>0, f is ….

concave up

When f’’(x)<0, f is…

concave down

When f’’(x)=0, f has a….

possible point of inflection

When f’’(x) changes signs, f has a…

confirmed point of inflection

When f’(x) changes from positive to negative, f has a….

maximum value

When f’(x) changes from negative to positive, f has a….

minimum value

What is the derivative of lnx?

1/x

What is the derivative of √x?

1/2√x

What is the derivative of a^x?

a^x X f’(x) X lna

What is the derivative of sin(x)?

cos(x)

What is the derivative of cos(x)?

-sin(x)

What is the derivative of csc(x)?

-csc(x)cot(x)

What is the derivative of tan(x)?

sec²(x)

What is the derivative of cot(x)?

-csc²(x)

What is the derivative of sec(x)?

sec(x)tan(x)

What is the derivative of arcsin(x)?

1/√(1-x²)

What is the derivative of arccos(x)?

-1/√(1-x²)

What is the derivative of arctan(x)?

1/(1+x²)

What are the requirements for the IVT?

f is continuous on the given closed interval and there exists a value ( c ) between endpoints (a,b) such that f( c)=L

What does the IVT do?

confirms that a continuous function must cross every y-value between endpoints; use to confirm that a value f(c ) = L

What is the EVT?

extreem value theorum

What does the EVT do?

confirms that if a function is continuous, it must have a maximum and minimum value

What are the requirements for EVT?

f is continuous on the given closed interval

What do you do when applying the EVT?

confirm that the function is continuous; find f’(x) and the critical points associated with it; plug the end points of the given interval and the critical points found back into the function to determine which has the minimum and maximum value (C-Test)

What are the requirements for MVT?

f is continuous on a closed interval; f is differentiable on an open interval; IROC=AROC

What does MVT confirm?

that there is at least one value where the IROC is equal to the AROC

How do you use MVT to solve problems?

set f’(x) equal to the average rate of change over the given interval and solve