Calculus 1: Graphing and Critical Points
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All Calculus 1LimitsDefinition of the DerivativeProduct and Quotient RulePower Rule and Basic DerivativesDerivatives of Trig FunctionsExponential and Logarithmic FunctionsChain RuleInverse and Hyperbolic Trig DerivativesImplicit DifferentiationRelated Rates ProblemsLogarithmic DifferentiationGraphing and Critical PointsOptimization ProblemsIndeterminate Forms and l'Hospital's RuleLinear Approximation and DifferentialsNewton Raphson MethodIndefinite IntegralsU SubstitutionDefinite Integrals and Fundamental TheoremApplications of Integration
Find the critical points of the function
##f(x) = \frac{5}{{(x^2 - 4x)}^2}##
##f(x) = \frac{5}{{(x^2 - 4x)}^2}##
For the following function draw a rough sketch of the graph for the 3 cases
##f(x) = \frac{(x - a)}{(x - b)(x - c)}##
Case 1: a < b, c
Case 2: b < a < c
Case 3: b, c < a
##f(x) = \frac{(x - a)}{(x - b)(x - c)}##
Case 1: a < b, c
Case 2: b < a < c
Case 3: b, c < a
Find and classify the critical points of the following function
##f(x) = x^3 - 3x^2 - 9x + 2##
##f(x) = x^3 - 3x^2 - 9x + 2##
Find the critical points of the following function
##f(x) = 6x^5 + 33x^4 - 30x^3 + 100##
##f(x) = 6x^5 + 33x^4 - 30x^3 + 100##
Find the critical numbers for the following function
##f(x) = \ln{(x^{2})} + 1.5x##
##f(x) = \ln{(x^{2})} + 1.5x##
Find the critical numbers of the function
##f(x) = x^2 - 4x##
##f(x) = x^2 - 4x##
Find the critical numbers for the following function ##f(x) = x^{\frac{2}{3}}##
Draw a rough graph of the following function using the critical numbers
##f(x) = \frac{2}{3}x^3 + \frac{9}{2}x^2 - 5x - 17##
##f(x) = \frac{2}{3}x^3 + \frac{9}{2}x^2 - 5x - 17##
Find the inflection points for the following function and determine intervals of concave up and concave down on the graph
##f(x) = \ln{(1 - \ln{(x)})}##
##f(x) = \ln{(1 - \ln{(x)})}##
Find the inflection points and intervals of concavity for the following function
##y = \frac{x^2 + 1}{x^2}##
##y = \frac{x^2 + 1}{x^2}##
Use the 2nd Derivative Test to find the inflection points and intervals of concavity for the following function
##f(x) = x^{3}(x - 4)##
##f(x) = x^{3}(x - 4)##
Find all points of inflection and discuss the concavity over different intervals for the following function
##f(x) = x^3 - 6x^2 + 12x##
##f(x) = x^3 - 6x^2 + 12x##
Use the second derivative test to find all relative extrema of the following function
##f(x) = \frac{1}{3}x^3 + 2x^2 + 3x##
##f(x) = \frac{1}{3}x^3 + 2x^2 + 3x##
Find the relative extrema for the following function on the given interval
##f(x) = \frac{2x}{x^2 + 1}## , ##[-2, 2]##
##f(x) = \frac{2x}{x^2 + 1}## , ##[-2, 2]##
Find the absolute extrema of the following function on the given interval
##h(x) = \frac{x}{x - 2}## , ##[3, 5]##
##h(x) = \frac{x}{x - 2}## , ##[3, 5]##
Find all relative maximum and minimum values for the following function
##f(x) = x^4 - 32x^2 + 256##
##f(x) = x^4 - 32x^2 + 256##