Multivariable calculus

Marginal rate of substitution

Differentiation rules

Functions and mappings

Second derivative

Factorization

Running head: DECISION-MAKING IN AGRIBUSINESS 1 DECISION-MAKING IN AGRIBUSINESS 8 Decision-Making in Agribusiness: Quantitative Applications (AGSC5300) Name Institution Decision-Making in Agribusiness: Quantitative Applications (AGSC5300) Problem 1 To differentiate the function, Y=?lnX?, the chain rule is applied. Fundamentally, the rule states that, for a function of the form Y=f(f(X)) the derivative is computed as follows: First, let f(X) be a value, say U. The function may therefore be written as Y=f(U) Then, the function is differentiated with respect to U That is, Next, the derivative of U with respect to X is calculated That is, Combining the above two derivatives gives the chain rule: =* Applying the above rule, let X? be U Therefore, Y=?lnU.

bccalcclassact5bparametric

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Analytic geometry

Differential geometry

Curves

Generalizations of the derivative

Tangent

Implicit curve

Envelope

Name:________________________ Date:________________________ AP Calculus BC Class Activity 5b: Parametric Equations and Calculus Find the derivative for the following parametric equations: x = t2 + 1 and y = 2t3 ? t2 x = 3 sin t and y = 4cos t ? 1 Find the second derivative for the following parametric equations: x = t2and y = t2 + 6t + 5 x = ln t and y = t2 + t A curve C is defined by the parametric equations x = t2 + t ? 1 and y = . Find in terms of t. Find an equation of the tangent line to C through the point where t = 2. A curve C is defined by parametric equations x = t2 ? t + 1 and y = t3 ? 3t. Find all points where the tangent line is horizontal. Find all points of vertical tangency.

bccalcclassact5aparametric

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Curves

Functions and mappings

Analytic geometry

equations

Name:________________________ Date:________________________ AP Calculus BC Class Activity 5a: Parametric Equations Complete the value of table below and graph the curve for and y = t x y -2 -1 0 1 2 3 Complete the table below and graph the curve for and . t x y 0 ? 2? For the following problems eliminate the parameter and find y as a function of x. and and Given the function below and an equation for x, find one parametric curve that could represent it: and and Use

bccalcclassact3aextrema

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Mathematical optimization

Maxima and minima

Sample maximum and minimum

Critical point

Interval

Name:________________________ Date:________________________ AP Calculus BC Class Activity 3a: Extrema Find the critical points for the following functions: 4. Given , find the absolute maximum and absolute minimum on the interval . 5. Given , find the absolute maximum and absolute minimum on the interval . 6. Find and show the local maximums and local minimums for the function 7. Find the interval(s) for which the following function is increasing: 8. Find and show the local minimums and local maximums for given
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Calculus AB #3

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Mathematics

?? ?? ? ? ?? ?? ? ? ?? ?? ? ? ?? ?? ? ? Ws 2 Calculus AB Harter 1. Identify each as odd, even or neither. a. 3 4 4y x x? ? ? b. c. 4 2y x? ? d. 2. Find the equation of a line in the form indicated that a. Passes thru (-2,-5) and (-5,3) (standard form) b. Perpendicular to 2x-3y=4 and passes thru (3,-3) (point slope form) 3. Solve the inequality. Write your answer in interval notation. a. 2 5 6x? ? ? b. 3 2 2x ? ? ? c. 2 3 4 8x? ? ? ? d. 3 22 2 3 3x x x? ? ? e. 2 6 8x x? ? ? 4. Convert each radian measure to degrees. a. 3 2 ? b. 5 6 ? c. 2 9 ? d. 4 3 ? e. 3 10 ? f. 19 180 ? g. 5 18 ? h. 11 6 ? i. 3 5 ? 5. Convert each degree to a radian measure. a. 210

Calculus AB #2

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Mathematics

Elementary algebra

equations

Ordinary differential equation

Linear equation

Function

Curve sketching

education

?3 ?2 ?1 1 2 3 4 16 ?12 ?8 ?4 4 8 12 Ws 1 ? review intercepts, linear equations, inequalities, symmetry, domain/range Calculus AB SHOW WORK IN YOUR NOTEBOOK. Find the x and y intercepts of each. 1. 1 4 2 y x= ? 2. 2 3 6x y? = 3. 3 24 4 15y x x x= + ? 4. 22 11 6y x x= + ? Name the domain and range of each. Write your answers in interval notation. 5. ( ) 1 3 f x x = ? 6. 3y x= + 7. 236y x= ? 8. ( ) 3xg x x += Linear equations 9. Find the equation of the line that passes through the points (2,-3) and (-3,-1) in standard form. 10. Find the equation of the line that passes through the points (-3,-3) and (5,0) in point slope form. 11. Find the equation of the line that passes through the points (2,-5) and (2,-6) in any form.

Calculus AB #1

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Mathematics

Analytic geometry

Elementary mathematics

Slope