3.2 Differentiation Rulesrepository.uobabylon.edu.iq/mirror/resources/paper_1_16616_476.pdf ·...
Transcript of 3.2 Differentiation Rulesrepository.uobabylon.edu.iq/mirror/resources/paper_1_16616_476.pdf ·...
-
College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim
49
The Intermediate Value Property of Derivatives
3.2 Differentiation Rules
Ex:- = → =
Ex:- = → =
Ex:- = → = ( ) = ( ) = ( ) =
-
College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim
50
Ex:- ( ) = + → = 5 + 2Products and Quotients
Ex:- For = ( + )( + + ) find a) by applying the product rule, and b) by multiplying the factors to produce a sum of simpler terms to differentiate.Solution:-
Ex:-find for:- 1) = +− 2) = ( − )( + + )Solution:-1)
2)
-
College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim
51
Negative Integer Powers of x
Ex:-
Second- and Higher-Order Derivatives
-
College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim
52
Ex:- = ( − ) + + find and .Solution:-
= ( − ) + + = + + − − − = − = −= − (− ) =
= (− ) = −
Ex:- Find the derivative of all orders of the function =Solution:-
3.3 The Derivative as a Rate of ChangeInstantaneous Rates of Change
Motion Along a Line: Displacement, Velocity, Speed, Acceleration, and Jerk
figure below) is
-
College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim
53
Ex:- For ( ) = − + of a body moving on a coordinate line, with s in meters and t in seconds.
a. Find the body’s displacement and average velocity for ≤ ≤ .b. Find the body’s speed and acceleration at the endpoints of the interval.c. When, if ever, during the interval does the body change direction?
Solution:-
-
College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim
54
Ex:-
Solution:-
3.4 Derivatives of Trigonometric Functions
Ex:- find the derivative of ( ) =
Ex:-
-
College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim
55
3.5 The Chain Rule and Parametric Equations
Ex:- in
1. 2.
Solutions:-
1.
2.
“Outside-Inside” Rule
Ex
Ex
Ex
-
College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim
56
Parametric Equations
Slopes of Parametrized Curves
-
College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim
57
Ex:- For the given parametric equations and parameter intervals for the motion of a particle in
the xy-plane, identify the particle’s path by finding a Cartesian equation for it and then graph
the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction
of motion.
1.
2.
3.
Solution:-
1. 2.
3.
Ex:- Find a parametrization for line segment with endpoints (-1,3) and (3,-2).
Solution:-
Ex:- For
Solution:-
-
College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim
58
3.6 Implicit Differentiation
Implicit Differentiation
Rational Powers of Differentiable Functions
Ex:-
Ex:- for:- 1) 2)Solution:-1.
2.
Ex:- for .
Solution:-
Ex:- For verify that the point (-1,3) is on the curve and find the lines that are:-(a) tangent and (b) normal to the curve at the given point.
Solution:-
3.7 Related Rates
-
College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim
59
Solution:-
Ex:-
Solution:-
Ex:-
Solution:-
Ex:-