12th grade calculus

Example

f(x) = (x² − 1)/(x − 1). It has no value at x = 1 itself. x: 0.9 | 0.99 | 0.999 | 1.001 | 1.01 | 1.1 f(x): 1.9 | 1.99 | 1.999 | 2.001 | 2.01 | 2.1 Estimate lim(x→1) f(x).

1. Both sides approach 2.
2. 2

Answer: 2

Example

Find lim(x→−3) (3x² + x − 4).

1. 3 × (−3)² + 1 × (−3) − 4 = 20

Answer: 20

Example

Find the limit as x → ∞ of (x² + 2x + 3)/(2x² + 7x − 1). Give a fraction if needed.

1. 1x²/2x² → 1/2

Answer: 1/2

Example

For y = x², find the gradient of the chord from x = −3 to x = −2.9.

1. (8.41 − 9) ÷ 0.1
2. = −5.9

Answer: -5.9

Example

For f(x) = x² − 4x, simplify (f(x + h) − f(x))/h.

1. (2xh + h² − 4h)/h
2. 2x + h − 4

Answer: 2x + h − 4

Example

Differentiate y = −4x².

1. dy/dx = -4 × 2x
2. = −8x

Answer: -8*x^1

Example

Find dy/dx when y = −3x³ − x² − 3x + 6.

1. dy/dx = −9x² − 2x − 3

Answer: -9x^2-2x-3

Example

Differentiate y = −2/x³.

1. y = −2x^(−3)
2. dy/dx = 6x^(−4) = 6/x⁴

Answer: 6/x^4

Example

Find dy/dx when y = (x − 6)(x − 1).

1. y = x² − 7x + 6
2. dy/dx = 2x − 7

Answer: 2x-7

Example

Find the gradient of y = 2x³ − 2x² − 6x + 1 at x = 0.

1. dy/dx = 6x² − 4x − 6
2. = −6

Answer: -6

Example

Find the equation of the tangent to y = −x² − 5x + 4 at x = −3. Type the right-hand side of y = … .

1. point (−3, 10), m = 1
2. y = x + 13

Answer: x+13

Example

Use calculus to find the stationary point of y = x² − 10x + 32. Type it like (2, −1).

1. dy/dx = 2x − 10 = 0 → x = 5
2. y = 7
3. (5, 7)

Answer: (5, 7)

Example

For which values of x is f(x) = −x² − 12x + 3 increasing?

1. x < −6

Answer: x < −6

Example

Find d²y/dx² when y = 2x⁴ + 4x³ + 5x² + x + 5.

1. dy/dx = 8x³ + 12x² + 10x + 1
2. d²y/dx² = 24x² + 24x + 10

Answer: 24x^2+24x+10

Example

A farmer has 80 m of fencing to make a rectangular pen against a straight wall (the wall is one side). If the two sides at right angles to the wall are x m, the area is A = x(80 − 2x). Find the maximum area in m².

1. A = 80x − 2x²
2. dA/dx = 80 − 4x = 0 → x = 20
3. A = 20 × 40 = 800 m²

Answer: 800

Example

A particle moves so that its displacement is s = t³ + t² + 9t metres after t seconds. Find its velocity at t = 5.

1. v = 3t² + 2t + 9
2. v(5) = 94 m/s

Answer: 94

Example

Find ∫ 20x³ dx. Leave out the + C.

1. 20x⁴/4
2. = 5x⁴ + C

Answer: 5*x^4

Example

Evaluate ∫ from 0 to 2 of (6x² − 3) dx.

1. [2x³ − 3x] from 0 to 2
2. 10 − 0 = 10

Answer: 10

Example

Find the area between y = x² + 2, the x-axis, x = 0 and x = 6.

1. [x³/3 + 2x] from 0 to 6
2. = 84

Answer: 84

Example

dy/dx = −4x − 5 and the curve passes through (3, 4). Find y in terms of x.

1. y = −2x² − 5x + C
2. 4 = −33 + C → C = 37
3. y = −2x² − 5x + 37

Answer: -2x^2-5x+37

Example

V(t) is the number of users of an app after t days. V′(15) = 32. What does this mean?

1. On day 15, users are increasing at about 32 per day

Answer: On day 15, users are increasing at about 32 per day