Derivatives and Integrals Basic Rules AP Calculus AB (copy)

(d/dx) sin x

cos x

(d/dx) cos x

-sin x

(d/dx) tan x

sec²x

(d/dx) cotx

-csc²x

(d/dx) sec x

sec x tan x

(d/dx) csc x

-csc x cot x

(d/dx) xⁿ

nxⁿ⁻¹

(d/dx) ln x

1/x

(d/dx) e^x

e^x

(d/dx) a^x

a^x ln a

(d/dx) log_a(x)

1/(x multiplied by ln(a))

(d/dx) [f(x)g(x)]

f′(x)g(x) + f(x)g′(x)

(d/dx) [f(x)/g(x)]

f′(x)g(x)−f(x)g′(x)​
[g(x)]²

f(x) is increasing

f'(x) > 0

f(x) is decreasing

f'(x) < 0

f(x) has a relative maximum or minimum

f'(x) = 0 or is undefined at a critical point

f(x) has a point of inflection

concavity changes

f(x) is concave up

f''(x) > 0

f(x) is concave down

f''(x) < 0

f(x) has a point of inflection

f''(x) changes signs

f'(x) is increasing

f''(x) > 0

f'(x) is decreasing

f''(x) < 0

f'(x) has a relative maximum or minimum

f''(x) = 0 or is undefined at a critical point

f'(x) has a point of inflection

f ''(x) = 0 and must change signs

f'(x) changes from negative to positive

f(x) has a relative minimum

f'(x) changes from positive to negative

f(x) has a relative maximum

f'(x) has a relative maximum or minimum

f(x) changes between increasing and decreasing

f''(x) changes from positive to negative

f'(x) changes from increasing to decreasing

f''(x) changes from negative to positive

f'(x) changes from decreasing to increasing

f(x)

At [a=x with units], the [context] is [c with units].

f^-1(c)=a

The [context] reaches [c with units] at [x=a with units]

f'(a)=k

At [x=a with units], The [context] is [increasing or decreasing] at a rate of [k with units of measure].

When is the particle moving at a constant speed? (Graph #1)

0 < t < 1 (where the graph has a horizontal segment)

When is the particle moving to the right? (Graph #2)

0 < t < 2 (when velocity v(t) > 0 / above the x-axis)

When is the particle moving to the left? (Graph #3)

2 < t < 5 (when velocity v(t) < 0 / below the x-axis)

When is the particle speeding up? (Graph #4)

2 < t < 3 (when v(t) and a(t) have the same signs)

When is the particle slowing down? (Graph #5)

1 < t < 2 and 3 < t < 5 (when v(t) and a(t) have opposite signs)

When is the velocity increasing? (Graph #6)

3 < t < 5 (when acceleration > 0 / positive slope)

When is the velocity decreasing? (Graph #7)

1 < t < 2 and 2 < t < 3 (when acceleration < 0 / negative slope)

Are the answers to velocity increasing/decreasing the same as speeding up/slowing down? (Graph #8)

No; velocity increases when acceleration is positive, whereas speeding up depends on velocity and acceleration having the same sign.

How fast is the particle moving at time t = 4, and in what direction? (Graph #9)

5 ft/sec moving left (|v(4)| = |-5| = 5 ft/sec; below x-axis)

What is the particle's velocity at time t = 4? (Graph #10)

-5 ft/sec (v(4) = -5 ft/sec)

What is the average velocity over the time interval 2 < t < 3? (Graph #11)

-5 ft/sec (displacement / time)

When does the particle change direction? (Graph #12)

t = 2 sec (at the x-intercept where v(t) changes sign)

How far does the particle move during the first second and which way? (Graph #13)

5 feet to the right (Area = 5 above the x-axis)

How far to the right does the particle go and when does it arrive there? (Graph #14)

7.5 feet to the right at t = 2 seconds (Area = 5 + 2.5 = 7.5)

What is the total distance traveled by the particle over 5 seconds? (Graph #15)

22.5 feet (Sum of all absolute areas: 5 + 2.5 + 5 + 10)

What is the particle's position after 5 seconds? (Graph #16)

-7.5 feet (Displacement: 7.5 - 15 = -7.5 feet)

What is the average velocity of the particle over the 5 second time interval? (Graph #17)

-1.5 ft/sec (Displacement / Total time = (7.5 - 15) / 5)

Are total distance and position after 5 seconds the same? Explain. (Graph #18)

No; total distance adds all areas positively (22.5 ft), while position uses actual directional signs/displacement (-7.5 ft).

What is the average speed over the 5 second interval? (Graph #19)

4.5 ft/sec (Total distance / Total time = 22.5 / 5)

What is the particle's acceleration over the 1st second? What does it mean? (Graph #20)

a = 0 ft/sec²; acceleration is not changing, so velocity is constant.

Over what time interval(s) does the particle have positive acceleration? (Graph #21)

3 < t < 5 (positive slope / increasing velocity)

Over what time interval(s) does the particle have negative acceleration? (Graph #22)

1 < t < 2 and 2 < t < 3 (negative slope / decreasing velocity)

When is the acceleration undefined on the interval 0 < t < 5? (Graph #23)

t = 1, 2, 3 seconds (not differentiable at sharp corners)

State an interval of time when the particle has positive acceleration but is slowing down. (Graph #24)

3 < t < 5 (positive slope, but graph is below the x-axis)

State an interval when the acceleration is negative but the particle is speeding up. (Graph #25)

2 < t < 3 (negative slope, below the x-axis)

What is the particle's acceleration over the time interval (3, 4)? (Graph #26)

a = 5 ft/sec² (Slope m = (0 - (-10)) / (5 - 3) = 5)

What is the particle's average acceleration over the entire 5 second time interval? (Graph #27)

-1 ft/sec² (a(t) = [v(5) - v(0)] / (5 - 0) = (0 - 5) / 5)

Over what time interval is the particle's velocity decreasing at a rate of 5 feet/sec every second? (Graph #28)

1 < t < 2 (where slope m = -5)

Motion Matrix: v(t) > 0 and a(t) > 0

Speeding up

Motion Matrix: v(t) > 0 and a(t) < 0

Slowing down

Motion Matrix: v(t) > 0 and a(t) = 0

Constant velocity, moving right

Motion Matrix: v(t) < 0 and a(t) > 0

Slowing down

Motion Matrix: v(t) < 0 and a(t) < 0

Speeding up

Motion Matrix: v(t) < 0 and a(t) = 0

Constant velocity, moving left

Motion Matrix: v(t) = 0 and a(t) > 0

Stopped, accelerating up or right

Motion Matrix: v(t) = 0 and a(t) < 0

Stopped, accelerating down or left

Motion Matrix: v(t) = 0 and a(t) = 0

Stopped, no acceleration or velocity