Pure Flashcards
steps to find nature of stationary points
- find the derivative
- make it equal to zero
- solve to get stationary points
- find derivative of derivative
- sub in stationary points into second derivative
- positive = minimum point, negative = maximum point
vertical asymptote is found by
makind the denominator equal to 0
horizontal asymptote is found by
making the x value really large
When finding antiderivative, remember:
to add c (where c is any constant)
how to sketch a v-t graph given a d-t graph
velocity is the derivative of displacement
how to use calculus to find the turning point of a quadratic:
- find derivative of quadratic
- solve for x
- sub x into quadratic equation to find y
equation for sum to Infinity In a convergent series.
S∞ = a / (1 - r)
Fully describe the transformation that maps the curve with equation y = x² - 3 onto the curve with equation y = 2x² - 6
[3 marks]
Stretch;
Parallel to y-axis;
Scale factor 2;
b3 points through a circle
PQ is the diameter of the circle, since
angle subtended at circumference is 90°
The equation of a curve is y=4x²+8x+1.
The curve is stretched parallel to the x-axis with scale factor 2.
Find the equation of the new curve, giving your answer in the form y=ax²+bx+c , where a, b and c are integers to be determined. [2]
→
- to stretch parallel to x axis: divide each x term by 2
- y = 4 (x/2)² + 8 (x/2) + 1
- = x² + 4x + 1
you CANNOT
log a negative number;
derivative of csc x
-cosec(x)cot(x)
derivative of secx
sec(x)tan(x)
derivative of cotx
-csc²x
derivative of cos(x)
-sin(x)
derivative of tanx
sec²x
derivative of sec²x
2sec²xtanx
Derived identities using sin²x + cos²x = 1 :
dividing by cos²x
tan²x + 1 = sec²x
dividing by sin²x
1 + cot²x = cosec²x
How small does angle have to be in small angle approximations?
- angle should be measured in radians
- In small angle approximations, the angle (in radians) should be small enough that sin(θ) ≈ θ and cos(θ) ≈ 1
Typically, this is valid for angles less than about 0.1 radians (approximately 5.7 degrees)
radian to degrees:
π radians = 180°
1 radian = 180° / π
horizontal asymptotes:
- degree on top < degree on bottom then y = 0
- both same: take the highest powers and divide them eg. 2x^2 / 3x^2 = 2/3
