Mathematicians
Often considered the "father of Calculus". Found the speed of falling objects.
Who is Issac Newton?
Integration By Parts formula
∫u dv
∫u dv= uv -∫v du
∫x3+x2+4x+(1/x)+(1/(1+x2))dx
x4/4 +x3/3+2x2+ln(|x|)+tan-1(x)+C
Given x=t2 and y=t3-3t. Find values of t where the curve has horizontal tangent lines.
t=-1, 1
The 3 conditions for continuity
1. limx->c- =limx->c+
2. f(c) exists
3. limx->c= f(c)
German mathematician who developed infinitesimal calculus independently and introduced the integral sign notation
Who is Gottfried Wilhelm Leibniz?
Parametric area under curve using x(t)&y(t)
∫y(t)*x'(t) dt
∫x2√(x3+1)+x√(x+1) dx
What is (2(x3+1)3/2)/9+ (2(x+1)5/2)/5 - (2(x+1)3/2)/3 +C
Give the position of particle is defined by x(t)=√(t) and y(t)=(t2)/4 . What is the velocity when x=2.
Vel: <1/4, 2>
Find two numbers whose sum is 30 and whose product is as large as possible.
15, 15
Swiss mathematicians expanded on both Newton's and Leibniz's work and introduced function notation f(x). He gave the definition of the constant e and has a certain method named after him useful for estimating a solution of a differential equation
Who is Leonhard Euler?
derivative of tan-1(x)
1/(x2+1)
∫ln(x)+(1/(x2-3x+2)) dx
What is xlnx-x-ln|x-1|+ln|x-2|+C
Find the second derivative of the line given by the parametric equations x=ln|5t²| and y=2t³/3
(3/2)t³
A balloon rises at a rate of 4 meters per second from a point on the ground 50 meters from an observer. Find the rate of change of the angle of elevation of the balloon from the observer when the balloon is 50 meters above the ground.
What is 0.04? (1/25)
19th-century German mathematician formalized the concept of limits and continuity, and differentiability. Developed many convergence tests still used today.
Who is Karl Weierstrass
Power reducing formula for sin2(θ), used to integrate polar functions by hand. Commonly used in Chapter 9
(1-cos(2θ))/2
∫xtan2(x) dx
-x2+xtan(x)+(x2)/2 + ln|cos(x)|+C
Find the area of the region enclosed by r=3(1+cos(theta))
27pi/2 units squared
The definition of calculus, where did it stem from (what is the literal meaning of calculus)
Mathematical study of continuous change. In Latin, calculus means pebbles where they used small stones to trade and that's where the origin of calculus stemmed from