PCH7: The Tangent to a Curve & The Derivative of a Function

Question 1

Limits

Answer 1

limf(x)
x -> c

Sub c into f(x)

limf(x)
x -> ∞

make every value over x, simplify and cancel, any value still over x is then equal to 0

Question 2

Continuity

Answer 2

• A curve is continuous if it is unbroken
• A function is continuous at x = c if
  —> f(x) is defined at c
  —> limf(x) exists
  x –> c
  —> f(c) is equal to this limit

Question 3

What is a secant?

Answer 3

Straight line passing through two points on the curve

Question 4

What is the formula for a gradient of a secant

Answer 4

m = f(x) - f(c)
————-
x- c

Question 5

What is a tangent to a curve?

Answer 5

Line which touches the cure in just one point

Question 6

What is the formula for a gradient to a tangent (using limits)

Answer 6

f ‘(x) = f(x+h) - f(x)
x —> 0 —————-
h

Question 7

how do we differentiate from first principles?

Answer 7

f ‘(x) = f(x+h) - f(x)
h —> 0 —————-
h

Question 8

Derivative of x^n

Answer 8

f (x) = ax^n

f ‘ (x) = an x^n-1

Question 9

What is a normal to a curve?

Answer 9

A line perpendicular to the tangent at that point

Question 10

How do you find the gradient of a normal to the curve?

Answer 10

m1m2 = -1

then find gradient using first principles or x^n

Question 11

Chain rule

Answer 11

f(x) = (ax^n1 + bx)^n2

1. Bring power outside of bracket to the front of the bracket
2. Minus one from the power outside of the bracket
3. Multiply by derivative of contents of bracket

Question 12

Product rule

Answer 12

y = vu
y' = vu' + uv'

Question 13

Quotient rule

Answer 13

where the numerator = u and the denominator = v

y’ = (vu’ -uv’) ÷ v^2