# NHS AP Calculus

## 50 terms

1

0

### definition of derivative

lim (f(x +h) - f(x))/h
h ->0

### alternate definition of derivative

lim (f(x) - f(c))/(x -c)
x ->c

### symmetric difference quotient

(f(a +h) - f(a -h))/2h

cos x

-sinx

sec^2x

sec x tan x

-csc^2x

1 / x

1 /(x lna)

e^x

1/sqrt(1 - x^2)

1/(1 + x^2)

### deriv(inverse secx)

1/(abs(x)*sqrt(x^2-1)

a^x*lna

### deriv(inverse f(a))

1/f'(inversef(a))

### mean value theorem

f'(c) = (f(b) -f(a))/(b -a)
where f(x) is differentiable

### critical number

f'(c) = 0 or f'(c) is undefined

f'(x) > 0

f'(x) < 0

### 1st derivative test max

f'(x) > 0 for x < c and f'(x) < 0 for x > c

### 1st derivative test min

f'(x) < 0 for x < c and f'(x) > 0 for x > c

### f(x) is concave up

f''(x) > 0 OR f'(x) is increasing

### f(x) is concave down

f''(x) < 0 OR f'(x) is decreasing

### 2nd derivative test max

f'(c) = 0 and f''(c) < 0

### 2nd derivative test min

f'(c) = 0 and f''(c) > 0

### integral x ^ n dx

(x^(n +1))/(n +1) + C

-cos x + C

sin x + C

tan x + C

-cot x + C

sec x + C

-csc x + C

### even function: integral of f(x) dx from -a to a

2 * integral of f(x) dx from 0 to a

0

0

### average value

1/(b-a) integral of f(x) dx from a to b

f(g(x)) * g'(x)

ln abs(x) + C

e^x + C

a^x/lna + C

### integral du / sqrt(a^2 - u^2)

inverse sin (u / a) + C

### integral du / (u^2 + a^2)

(1/a) inverse tan (u / a) + C

### integral du/(u * sqrt(u^2 - a^2))

(1/a) inverse secant (abs(u)/a) + C

### exponential growth

A(t) = A0 * e^(kt)

### Newton's Law of Coolling

u(t) = T + (u0 -T)e^(kt)

### Average rate of change

(f(b) - f(a))/(b-a)

-csc x cot x