Calculus Study guides, Class notes & Summaries

What is the difference quotient? - ANSWER-f(x+h)-(f(x) _______________ h Lim f(x) = 4 x > 2 - ANSWER-As x approaches 2, y is very close to 4 Lim f(x) = 2 x > 2 - ANSWER-As x approaches 2, y is either undef. or = to 4 Does the limit exist? Lim f(x) = 4 x > 2 - ANSWER- Lim x^2 - 9 ________ x > 3 x - 3 - ANSWER-Step 1: (x-3)(x+3) _____________ x-3 Step 2: (cross out)(x+3) ______________________ cross out Step 3: x+3 Step 4: 3 What is the de...

Average Rate of Change - ANSWER-Slope of two points Instantaneous Rate of Change - ANSWER-Derivative Derivative is - ANSWER-the slope of the tangent line at one point Mean Value Theorem - ANSWER-If a function is both continuous and differentiable on an interval, then there exists some 'c' in that interval at which the average rate of change equals the instantaneous rate of change Normal line - ANSWER-Perpendicular to a tangent line Definition of Derivative - ANSWER-The limit as ...

What can be used to determine whether a relation satisfies the definition of a function? - ANSWER-Vertical Line Test What are the four ways a function can be represented? - ANSWER-numerically, algebraically, verbally, graphically What must be met in order to be considered a function (basic level here) - ANSWER-one output for every input A table of data is a ____ representation of a function - ANSWER-numerical A scatterplot is a ____ representation of a function - ANSWER-graphical ...

Constant Rule - ANSWER-d/dx [c] = 0 Constant Multiple Rule - ANSWER-d/dx [cf(x)]=cf'(x) Sum Rule - ANSWER-d/dx [f(x)+g(x)] = f'(x)+g'(x) Difference Rule - ANSWER-d/dx [f(x)-g(x)] = f'(x)-g'(x) Product Rule - ANSWER-d/dx [f(x)*g(x)] = f(x)g'(x) + f'(x)g(x) Quotient Rule - ANSWER-d/dx [f(x)/g(x)] = [g(x)f'(x) - g'(x)f(x)]/[g(x)²] Chain Rule - ANSWER-d/dx [f(g(x)] = f'(g(x))*g'(x) Power Rule - ANSWER-d/dx[xⁿ]=nxⁿ⁻¹ Definition of the number e - ANSWER-d/dx[...

sin²x + cos² x=? - ANSWER-1 Which Pythagorean formula = 1? 1 + tan² x=? - ANSWER-sec²(x) Which Pythagorean formula = sec²(x)? 1 + cot² x=? - ANSWER-csc² x Which Pythagorean formula = csc² x sin(-x)=? - ANSWER--sin x -sin(x) = sin(-x) or only -sin(x)? d/dx (x^n) Rule? - ANSWER-Power Rule Reverse the power rule for derivatives. d/dx (f*g) - ANSWER-fg' + gf' Product Rule What is the product rule in reverse? d/dx (f/g) Rule? - ANSWER-Quotient Rule When do you use t...

definition of a finite limit at a real number - ANSWER-and we say that "the limit of f(x) as x approaches c* is L" if we can make the values of f(x) as close to L as we want by taking x close enough to c on either side of c but not equal to c. definition of finite one-sided limits at a real number (left) - ANSWER-and we say that "the limit of f(x) as x approaches c from the left is L" if we can make the values of f(x) as close to L as we want by taking x close enough to c but less than c....

Linear Function - ANSWER-F(x)=mx+b Quadratic Function - ANSWER-F(x)=ax^2+bx+c Exponential Function - ANSWER-F(x)=ab^x Logarithmic Function - ANSWER-F(x)=Logbx=y or b^y=x Finding the slope of a line - ANSWER-y-y^1=m(x-x^1) Exponential Function in General Form - ANSWER-A(t)=Pb^t (Pb^t) The constant P represents - ANSWER-y-intercept (Pb^t) The constant b represents - ANSWER-value of y is multiplied by b for every one unit increase of t b>1 - ANSWER-exponential growth b...

Find the average rate of change of f(x) on [a,b] - ANSWER-find: [f(b)- f(a)]/ (b-a) Difference Quotient - ANSWER-f(x+h)-f(x)/h, where h does not equal 0. using difference quotient when f(x) is a radical - ANSWER-multiply top and bottom by conjugate using difference quotient when f(x) is a fraction - ANSWER-multiply top and bottom by common denominator of top fractions. Find instantaneous rate of change - ANSWER-use definition of derivative (difference quotient) as lim x-> 0 how...

A function is continuous at x=A when... - ANSWER-1.) f(A) exists 2.) The limit exists -the LHL and RHL are equal 3.) f(A)=limit Limit definition of the derivative - ANSWER-the limit as h approaches zero is [f(x+h)-f(x)]/h Fundamental Theorem of Calculus I - ANSWER-Derivatives and integrals are inverses derivative of sin - ANSWER-cos derivative of cos - ANSWER--sin derivative of tan - ANSWER-sec^2 derivative of cot - ANSWER-csc^2 derivative of sec - ANSWER-secxtanx ...

McLaurin Series for 1/(1-x) - ANSWER-∑xⁿ=1+x+x²+x³+x⁴+... R=1 McLaurin Series for e^x - ANSWER-∑xⁿ/n!=1+x/1!+x²/2!+x³/3!+... R=infinity McLaurin Series for sinx - ANSWER-∑(-1)ⁿ(x²ⁿ⁺¹)/(2n+1)!=x-x³/3!+x⁵/5!-x⁷/7!+... R=infinity McLaurin Series for cosx - ANSWER-∑(-1)ⁿ(x²ⁿ)/(2n)!=1-x²/2!+x⁴/4!-x⁶/6!+... R=infinity Conditionally Convergent - ANSWER-A series is ∑ a_n is called conditionally convergent if it is convergent but not absolutely c...