Calculus I for Science and Engineering

Calculus III for Engineering

Elementary Differential Equations

Derivative Tests, Curve Sketching Goal: Use information about the derivatives of a function f to obtain information about the graph of f itself. Theorem: Suppose f is continuous and diﬀerentiable on [ a, b ]. Then: Proof: First Derivative Test: Suppose c is a critical point of f and f is diﬀerentiable (except perhaps at c ). Then: Geometric Interpretation: T

MATH | 2250 | Calculus I for Science and Engineering | homework | Derivative Tests  Curve Sketch

Substitution Rule Goal: Develop a method for integrating certain compositions of functions. Then we can integrate even more! Theorem: (Substitution Rule) If u = g ( x ) is diﬀerentiable and f is continuous, then Examples: Evaluate the following indeﬁnite integrals. (1) Z √ 2 x +1 dx (2) Z x 3 cos( x 4 + 2) dx (3) Z x √ 1 - 4 x 2 dx (4) Z 4 cos( x ) e sin x dx (5) Z x 5 √ 1+ x 2 dx (6)

MATH | 2250 | Calculus I for Science and Engineering | homework | Substitution Rule Goal: Develo

Remarks: • The level curve f ( x, y )= c lies entirely in the domain of f , and on this curve f is constantly c . • The level curves f ( x, y )= c are just the traces of the graph of f in the horizontal plane z = c projected down to the xy -plane. • A collection of level curves, well drawn and labeled, can lead to a good view of the overall behavior of a function. For instance, the s

MATH | 2500 | Calculus III for Engineering | lecture notes | Remarks: • The level curve f (

13.6.pdf-Tangent Planes and Linear Approximations One

MATH2 700 , Summer 201 8 Name: Worksheet 4 Section: Due Wednesday, September 18th For full credit, you must show all work and box answers. 1. Solve the following diﬀerential equations or initial-value problems by separation of variables. Put your solutions in explicit form. (a) y 0 = e - y sin(2 x ) , y (0) = 0 (b) y 0 +2 xy =0 , y (0) = 1 (c) ( x + 2) dy dx = y 2 +2 2. Solve the f

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