connecting-f-with-the-graph-of-f-applications-of-derivative-4-3

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The student will investigate derivatives presented in graphic, numerical, and analytic

contexts and the relationship between continuity and differentiability. The derivative will

be defined as the limit of the difference quotient and interpreted as an instantaneous rate

of change.

Mathematics Standards of Learning

60

APC.6 The student will investigate the derivative at a point on a curve. This will include

a) finding the slope of a curve at a point, including points at which the tangent is vertical

and points at which there are no tangents;

b) using local linear approximation to find the slope of a tangent line to a curve at the

point;

c) defining instantaneous rate of change as the limit of average rate of change; and

d) approximating rate of change from graphs and tables of values.

APC.7 The student will analyze the derivative of a function as a function in itself. This will

include

a) comparing corresponding characteristics of the graphs of f, f ', and f ";

b) defining the relationship between the increasing and decreasing behavior of f and the

sign of f ';

c) translating verbal descriptions into equations involving derivatives and vice versa;

d) analyzing the geometric consequences of the Mean Value Theorem;

e) defining the relationship between the concavity of f and the sign of f "; and

f) identifying points of inflection as places where concavity changes and finding points

of inflection.

APC.8 The student will apply the derivative to solve problems. This will include

a) analysis of curves and the ideas of concavity and monotonicity;

b) optimization involving global and local extrema;

c) modeling of rates of change and related rates;

d) use of implicit differentiation to find the derivative of an inverse function;

e) interpretation of the derivative as a rate of change in applied contexts, including

velocity, speed, and acceleration; and

f) differentiation of nonlogarithmic functions, using the technique of logarithmic

differentiation. *

* AP Calculus BC will also apply the derivative to solve problems. This will include

a) analysis of planar curves given in parametric form, polar form, and vector form,

including velocity and acceleration vectors;

b) numerical solution of differential equations, using Euler's method;

c) l'Hopital's Rule to test the convergence of improper integrals and series; and

d) geometric interpretation of differential equations via slope fields and the relationship

between slope fields and the solution curves for the differential equations.

APC.9 The student will apply formulas to find derivatives. This will include

a) derivatives of algebraic, trigonometric, exponential, logarithmic, and inverse

trigonometric functions;

b) derivations of sums, products, quotients, inverses, and composites (chain rule) of

elementary functions;

c) derivatives of implicitly defined functions; and

d) higher order derivatives of algebraic, trigonometric, exponential, and logarithmic,

functions. *

* AP Calculus BC will also include finding derivatives of parametric, polar, and vector

functions.

Mathematics Standards of Learning

61

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