CALC 2250 Final exam questions and answers.

linear graph quadratic graph y=x^2 exponential graph square root/radical graph rational graph f(x)= 1/x absolute value graph continuity steps 1. plug x into f(x) and find answer 2. take limit as x--># and see if it is equal to (1) 3. if they are equal, then they are continuous chain rule f'(x)=f'(g(x)) x g'(x) d/dx (sin(x))= =cos(x) d/dx (cos(x))= =-sin(x) d/dx (tan(x))= =sec^2(x) d/dx (sec(x))= =sec(x)tan(x) d/dx (csc(x))= =-csc(x)cot(x) d/dx (cot(x))= =-csc^2(x) csc(x)= 1/sin(x) sec(x)= 1/csc(x) cot(x)= 1/tan(x)=cos(x)/sin(x) d/dx(arcsinx)= 1/sqrt(1-x^2) d/dx(arccosx)= -1/sqrt(1-x^2) d/dx(arctanx)= 1/(1+x^2) average velocity equation change in position/change in time OR (pos. at time b)-(pos. at time a)/(b-a) instantaneous velocity equation v(t)=lim t-->a (h(t)-h(a))/(t-a) displacement s(t) velocity v(t) acceleration a(t) s'(t)= v(t) v'(t)= a(t) When do you use log differentiation? when variable is in base and exponent relative extrema does not include... endpoints, only critical points if tangent line is below curve, then... concave up if tangent line is above curve, then... concave down closed interval method steps 1. find critical points using first deriv 2. evaluate f(x) at endpoints and critical points 3. largest of those is abs. max and smallest is abs. min increasing/decreasing test steps 1. find critical points using first deriv 2. evaluate f(x) at its critical points 3. sections where number is positive means graph is inc. and sections where number is negative means graph is dec. first derivative test steps 1. find critical points using first deriv 2. evaluate f(x) at critical points 3. pos. # = inc and neg. # = dec. 4. at those breaks are where relative extrema of f(x) are concavity test steps 1. find critical points using second derivative 2. evaluate f''(x) to see if each section is positive or negative 3. if positive, graph is concave up, and if negative, graph is concave down second derivative test steps 1. find critical points using first derivative 2. evaluate f''(x) to see if each section is positive or negative 3. if positive, graph is concave up, and if negative, graph is concave down inflection points are where... concave up/down changes when finding inverse of a graph, mirror over... y=x line optimization steps 1. picture! 2. variables (label pic) 3. constraint 4. function 5. goal 6. solve for 1 variable in constraint 7. plug that value for that variable into the function and simplify 8. domain 9. derivative of function 10. set=0 to solve for critical points 11. first deriv. test, second deriv. test, or closed interval method 12. statement! related rates steps 1. picture! 2. write what you're given and variables in pic 3. goal! 4. equation and take deriv. 5. plug given values in to deriv. and solve for unknown 6. take value for unknown and plug in to equation to solve for goal 7. statement! lim x-->+- infinity (1/x) = 0 to find vertical asymptote... set denominator = 0 to find horizontal asymptote... take limit and plug 0 into x and solve sum of n from k=1 (c) = c x n sum of n from k=1 (k^2) = n(n+1)(2n+1)/6