Calculus Flashcards
To understand differentiation and integration more effectively
Critical points are points in the curve where the slope is
zero
All critical points are not Local maxima|Local mininma, but all local maxima and local minima has a critical point why
Criticial points can also indicate inflection points
What does y = x2 gives as curve
A Parabolic Curve
What does y = x2 gives as curve
Similar to a Asymptode
What is Differentiation tells you
Slope of a tangent line
What does integration used for
To find AUC and average rate of change of moving objects
What are the ways to find slopes
If lines - y = mx+c
if curves - Differentiation (Slope of tangent line)
Slope of Secant line
What is a Secant Line
Secant Line connects two points in a curve
Slope of a Secant Line
y2 - y1/ x2 - x1
how do we find instantaneous rate of change in terms of calculus
Differentiation
how do we find instantaneous rate of change in terms of calculus
Integration
How do we find acceleration in terms of Calculus
second derivative of the function
What is actually a function
A relation between Y value and X value
What is the difference between f(x) and F(x)
f(x) - differentiation,
F(x) - integration
What is the difference between f’(x) and f^-1(x)
f’(x) = differentiation
f^-1(x) = Inverse function
What is the use of this f^-1(x)
gives you the value of x wrt to y
What is sine wave
Starts at 0, peaks at 90, bottoms at 270 and again peaks at 360
What is Cosine Wave
Starts at 90 peaks at 0, bottoms at 180 and again peaks at 270
f(sinx)
cosx
f(cosx)
-sinx
f(tanx)
sec2x
f(cotx)
-cosex^2x
f(secx)
secx tanx
f(cosecx)
-cosecx tanx
What is Power rule in Differentiation
n^x = nx^n-1
What is chain rule
dy/dx(f(g’(x)) = f’(g(x) * g’(x)
outside first inside second
differentiate f(x) g(x)
f’(x) g(x) +f(x) g’(x)
differentiate f(x)/g(x)
f’(x) g(x) +f(x) g’(x)/(g(x))^2
differentiate c
0
differentiate x
1
What is LCD
Multiply the numerator and denominator by a common number so that the denominators match for easy calculations
What happens when we have bounds of Secant line infinitely close to each other
This resembles Tangent Line
What is the limit x->2 when y = x2 means
it means the the value of x is approaching Ly close to 2 it can be 2.1, 2.001
What does Squeeze theorem
if b(x) lies between a(x) and c(x) and
differentiation of a(x) and b(x) = d.
then c(x) is also zero
What are continuous curve and non continuous cutves
Non continuous curve has a break/missing(cannot be drawn in a single hand)
Continuous can
What are continuous curve and non continuous cutves
Non continuous curve has a break/missing(cannot be drawn in a single hand)
Continuous can
Give an example for non continuous curves
Jump discontinuity, asymptodes
What is an odd function and and even function
Odd function is symmentric across either x axis or y axis even function is symmetric across origin
What is an odd function and and even function
Odd function is symmentric across either x axis or y axis even function is symmetric across origin
What is domain and bounds
Domains are the highest and lowest values of x-axis and bounds are the highest and lowest values of y axis
How do you find concavity of a curve
Second derivative of a function gives the distance from the critical point
What does it mean if the second derivative of a function is positive
It means that the curve is upwards
What does it mean if the second derivative of a function is negative
The curve is concave down
How do you find critical value by Extreme value theorem
According to mean value theorem
step 1 take the first derivative of the curve and equate it to zero to find x values
step 2
substitute X value and identify y value from the curve step 3
apply the lower and higher bounds to the equation these are the critical points of the curve
When is first derivative preferred over second derivative while calculating critical points
If the formula involves fractions taking second derivative leads to complications hence first derivative is used
What is jerk
It is the third derivative of velocity or second derivative of acceleration
What is Local Maxima and Global Maxima
local maxima is a peak in relavance to the nearer points, global maxima is the maxima of all the peaks in the curve
How can We interpret limits which respective to their values
Limits from left = Limit from Right => Then its a asymptode
Feauture of Vertical Asymptode
y reaches some number, x reaches towards infinity
Feauture of Horizontal Asymptode
x reaches some number, y reaches towards infinity
Roles Theorem
If f is continous function
f is differentiable on (a,b)
f(a) = f(b)
Mean Value Theorem
If f is continous function
f is differentiable on (a,b)
Slope is calculated by y2-y1/x2-x1
What is Anti Derivative Called as
Integration
What is the Use of Integration
To Area Under the Curve by summing up infinitely small number of distances
i=1 -> n ∑ C =
C
i=1 -> n ∑ i =
n(n+1)/2
i=1 -> n ∑ i^2 =
n(n+1)(2n+1)/6
i=1 -> n ∑ i^3 =
n^2(n+1)^2/2
What is Riemann Integral
Definite Integral
What is Definite Integral
Definite Integral Has the bounds
What is INdefinite Integral
Indefinite Integral Has no bounds
∫f(x)dx = F(x) + C
How to define Sine, Cos, Tan by trigonometry
Sin = opposite/Hypotenuse,
Cosine = Adjacent/Hypotenuse
Tan = Opposite/Hypotenuse
What is Newtons Law in calculus used For
Newton Law is used to point a point in x axis where y value is zero
What is Newtons law of Calculus
x(n+1) = xn - ( f(xn)/f’(xn) )
Steps in optimization procedure
Understand the question
Identify the optimisation equation and the constraints eqn
Now solve the constraints equation to get simpler equation
Substitute this into the optmisation equation and optimize it (ie derivative =0)
Solve it and substitute to constrain eqn and find both values
What is optimization in calculus
Optimization helps us to maximize or minimise a function
(Ie, finding local maxima or minima)
What is optimization in calculus
Optimization helps us to maximize or minimise a function
(Ie, finding local maxima or minima)
Integration of rate gives
Accumulation
