Antiderivative - Study guides, Class notes & Summaries

These notes are in a pdf format. - colour coded - step-by-step examples for each topic The notes cover the topic: Antiderivatives, Mathematical Induction, Binomial theorem, Definite integral & Riemann sums

Limit Definition of Derivative Mean Value Theorem If f(x) is defined and continuous on [a,b] and differentiable on (a,b) then there is at least one point where f'(c) = f(b)-f(a)/b-a A function is continuous at a point x=a if: a) f(a) is defined b) Limit as x approaches a of f(x) c) Limit as x approaches a of f(x) is equal to f(a) Fundamental Theorem of Calculus (P1) If f is continuous on [a,b] F(x) = ∫f(x) and both are continuous on [a,b] F(x) is differentiable on ...

ALEKS MATH PLACEMENT WITH COMPLETE SOLUTIONS 100% 2023/2024 graph ≥, ≤ solid line graph >, < DOTTED line every, each, per (writing equations) rate/ set y value equal to (multiply this value by the x variable) When writing an equation or working ANY inequality problem what does it help to do first? write out what's happening What should you double check for EVERY time writing problem down wrong (NEGATIVE SIGNS), uncarried negative signs, unflipped inequality s...

graph ≥, ≤ solid line graph >, < DOTTED line every, each, per (writing equations) rate/ set y value equal to (multiply this value by the x variable) When writing an equation or working ANY inequality problem what does it help to do first? write out what's happening What should you double check for EVERY time writing problem down wrong (NEGATIVE SIGNS), uncarried negative signs, unflipped inequality signs, undotted lines, incorrect multiplication, fully...

The slope of a secant line represents what kind of change? - ANSWER-Average roc The slope of a tangent line represents what kind of change? - ANSWER-Instantaneous roc What kind of change does an integral represent? - ANSWER-Total change Explain what a derivative is graphically. - ANSWER-Slope of tangent line Explain what a derivative is numerically. - ANSWER-Difference quotient from a table Explain what a derivative is analytically. - ANSWER-lim of (f(x+h)-f(x))/h as h->0 3 ...

Vertical Asymptote - ANSWER-Set Denominator equal to zero Horizontal asymptote - ANSWER-set numerator equal to zero Rolles theorem states there is... - ANSWER-a tangent line at some point on a closed interval [a,b] if the following hold true a) continuous b) differentiable c) f(a)=f(b) Integration is a... - ANSWER-continuous addition process Intermediate Value Theorem: Suppose f is a function continuous at every point on the interval [a,b]: - ANSWER-a) f will take on every value b...

Integral of cosx - ANSWER-sinx+c Integral of sin x - ANSWER--cosx+c Integral of sec^2x - ANSWER-tanx+c Integral of secxtanx - ANSWER-secx+c Integral of csc^2x - ANSWER--cotx+c Integral of cscxcotx - ANSWER--cscx+c If a trapezoidal sum overapproximates ∫0 to 4 f(x)dx, and a right Riemann sum underapproximates ∫0 to 4 f(x)dx, which of the following could be the graph of y=f(x)? - ANSWER-A- The one with a y intercept at three that is concave up and decreasing Let f be the f...

The derivative of e9x is equal to e9x. - correct answer false The derivative of ex f(x) is equal to ex [f(x) + f '(x)] - correct answer true ln(54) − ln(12) is equal to ln(9) − ln(2). - correct answer true ln(ab) = ln(a) + ln(b) for all a,b > 0 - correct answer true /x^ndx=(1/n+1)x^n+1 + C for all n - correct answer ...

evaluate the integral 2/1, ln(x)/x dx - answer-(ln2)^2/2 consider the function f(x)=x^4-4x^3. which of the following is NOT true - answer-f has a local extrema at x=0 and x=3 find the work done by a force F=sqroot 2x+1 N to move an object along a straight path from a distance of 0 meters to a distance of 4 meters - answer-26/3 J an object is moving along the x-axis. Let s(t), v(t) and a(t) be functions giving the position, the velocity, and the acceleration of the object at time t. whic...

extreme value theorem - Correct Answer if f(x) is continuous on [a,b] then f(x) contains an absolute max and absolute min on [a,b] Fermat's theorem - Correct Answer if f(x) is differentiable on (a,b) and has a local max or min at c within (a,b), then f'(c) = 0 critical points - Correct Answer for f(x), point a at which either f'(a) = 0 or f is not differentiable at a (closed intervals include endpoints) mean value theorem - Correct Answer if f(x) is continuous on [a,b] and differenti...