Using Implicit Differentiation (Often with Respect to Time)

Math 1261 Calculus I
Test 3 Review Sheet
Sections 3.5, 3.6, 3.9, 3.10, 4.1-4.5

Related Rates (3.9)

Using implicit differentiation (often with respect to time)

Differentials and linear approximation (3.10)

Be able to approximate the values of y = f(x) near x = a with the tangent line

Understand the geometry of differentials

Know the definition of the differentials dy and dx for the function y = f(x) near x = a.

Use differentials to approximation function values such as sin(29°), (4.01)2, (103)1/2

Use differentials to approximate the difference in values of a function.

Be able to state the following definitions

function, limit, continuous at a point, derivative

differentials

the number e in terms of a limit

absolute maximum, absolute minimum, absolute extremum

relative maximum, relative minimum, relative extremum

a function f is increasing on an interval I

a function f is decreasing on an interval I

a function f is constant on an interval I

a function f is concave up on an interval I

a function f is concave down on an interval I

inflection points

(first order) critical number (CN) of a function f

second order critical number (CN2) of a function f

linear asymptotes

horizontal asymptote (HA) y = k where k is a real number

vertical asymptote (VA) x = h where h is a real number

oblique (slant) asymptote (OA) y = mx+b where m is a nonzero real number, b is a real number

Be able to state the following theorems

Extreme Value Theorem (affectionately known as the EVT)

Mean Value Theorem (affectionately known as the MVT); Rolle's Theorem

(Geometric Interpretations; Illustrate with examples why assumptions are necessary)

Fermat's Theorem

L’Hospital’s Rule

Be able to prove one part ofthe following theorem

If a function F is continuous on the closed interval [a,b] and
(1) F'(x) > 0 for all x in (a,b) then F is increasing on [a,b]
(2) F'(x) = 0 for all x in (a,b) then F is constant on [a,b]
(3) F'(x) < 0 for all x in (a,b) then F is decreasing on [a,b]

(OVER)

Be able to use (apply) the derivative rules to take derivatives including

§  Constant Rule; Constant Multiple Rule; Sum, Difference, Product, Quotient, and Chain Rules

§  Derivative of power functions for n a real number.

§  Derivatives of exponential functions for positive real numbers a.

§  Derivatives of trigonometric and inverse trigonometric functions

¨  sine, cosine, tangent, cotangent, secant, cosecant (3.3)

¨  inverse sine, inverse tangent, inverse secant (3.5)

¨  inverse cosine, inverse cosecant, inverse cotangent (3.5)

§  Derivatives of logarithmic functions for real numbers a > 0, a ≠1

§  perform implicit differentiation (3.5)

¨  and know the underlying assumption being made (3.5)

§  take higher order (2nd derivatives, 3rd derivatives, etc.) derivatives and use proper notation

§  perform logarithmic differentiation (3.6)

§  The number e (definition; e is expressible as a limit; connection to natural logarithm)

Graphing Functions using Derivatives (4.1-4.3, 4.5)

maximum and minimum values (relative and absolute); extremum, extrema

using derivatives to graph

first derivative: critical numbers (CNs), local extrema, increasing/decreasing/constant

second derivative: 2nd order critical numbers (CN2s), inflection points, concave up/down

theoretical basis: The Mean Value Theorem (MVT)

linear asymptotes

horizontal asymptotes y = L (limit at plus or minus infinity of the function is L)

vertical asymptotes x = c (limit at c from the left or right is plus/minus infinity)

oblique/slant asymptote y = mx+b (m not zero) <4.5>

As x goes to plus or minus infinity then F(x) - (mx+b) goes to zero

Graphing Functions given information about them

Section 4.3 problems 24-29 are examples of this

L’Hospital’s Rule (4.4)

Indeterminate forms

Applying L’Hospital’s Rule to evaluate limits when appropriate