# Course outline, MA 113, Spring 2014 Part A, Functions and limits Functions, domain and ranges, A Review (9 problems)

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1 Course outline, MA 113, Spring 2014 Part A, Functions and limits Functions, domain and ranges, A Review (9 problems) Functions, domain and range Domain and range of rational and algebraic functions Linear and quadratic functions Common models, distance traveled, geometric problems Find domain and range of a given function Find a formula given a description in words Find a linear function given two points or a point and a slope 1.3,1.5 Functions and inverse functions, A1.3and1.5-Functions (12 problems) Definition of composite function Definition of inverse functions Relation between graph of a function and its inverse Finding a formula for an inverse function Relation between domain and range for a function and its inverse Find inverse and composite fucntions Find domain and range of inverse functions Graph an inverse function 1.4,1.5 Trigonometric functions, A Trig (11 problems) Radian measure Definitions of sin and cos by unit circle Definitions of tan, sec, csc, and cot Pythagorean identities, addition formulae for sine and cosine, double angle formulae

2 Definitions of inverse trig functions Use of trigonometric functions to describe the relations between sides and angles in a right triangle Using identities and the unit circle to find values of trig functions Finding sides and angles in right triangles Evaluating expressions such as cos(arcsin(x)) Using cos(x) and sin(x) to describe points on a circle 1.6 Exponential and logarithm functions, A1.6-Exp-Log (12 problems) Definition of the logarithm with base e as the inverse of e x Properties of logarithms Write b x in terms of e ( x ln(b)) Solving equations involving exponential functions We do not cover hyperbolic functions in this course Using properties of logarithms to evaluate expressions involving logarithms and exponential functions Solving equations involving the logarithm and exponential function Finding A and k so that f(t) = Ae kt has specified properties Tangent and velocity problem, A Limits (15 problems) Slopes of secant lines, Average velocity Using numerical estimates to guess slope of tangent line and instantaneous velocity. Relation between one-sided and two sided limits, limits from a graph Compute average rate of change, compute average velocity

3 Write instantaneous velocity and slope of a tangent line as a limit. Finding limits from graphs Using one-sided limits to determine if a (two-sided) limit exists Guessing limits from numerical values or table of a function 2.3 Limit laws/computing limits, A2.3-Computing-Limits (9 problems) Limit rules for sums, products, quotients, and powers. Evaluating limits using these laws Providing step-by-step justifications for evaluating limits and recognizing when a rule does not apply 2.4 Limits and Continuity, A2.4-Continuity (15 problems) Definition of continuous functions Continuity of composite functions Evaluating limits of continuous functions by substitution. Finding the set where a function is continuous Using continuity to evaluate limits and recognizing when this does not apply Choosing parameters in piecewise functions to guarantee continuity 2.5 Evaluating limits algebraically, A2.5-Evaluating limits Indeterminate forms Simplify and use limit laws Computing slope of tangent line Distinguishing between limits which do not exist and limits where the rule for the limit of a quotient does not apply.

4 Evaluating limits where algebraic simplification is needed prior to applying limit laws Evaluating limits that arise in computing instantaneous velocities and slopes of tangent lines 2.6 Trigonometric Limits, A2.6-Trig-limits Squeeze theorem Limits of sin(x)/x, (1 cos(x))/x 2 Algebraic manipulations and trig limits. Note that 2.7 on limits at infinity will be covered later. State squeeze theorem. Provide a complete statement including hypotheses and conclusion Use squeeze theorem to evaluate limits. Recognize when the squeeze theorem does not apply.

5 Part B, The derivative 3.1 The derivative, B3.1-derivative (12 problems) Definition of the derivative Equation of tangent line Derivative of linear functions Approximating derivatives Computing derivative from definition Finding a tangent line Finding or estimating derivative from a graph 3.2 The derivative as a function, B3.2-Deriv-Function (15 problems) Power rule Derivative of e x and characterization of e by its derivative. Leibniz notation Continuity of differentiable functions Computing derivatives Finding tangent lines Proving a differentiable function is continuous Differentiability of piecewise functions 3.3 Product and quotient rules, B3.3-Product (11 problems) Proof of product rule Use of product rule and quotient rules Compute derivatives with the product rule State product rule 3.4 Rates of change 3.4, B3.4-Rates (13 problems)

6 Interpretation of derivative of as a marginal rate Speed, velocity and acceleration Motion under the influence of gravity Interpretation of derivative as a marginal rate Describing motion 3.5 Higher derivatives, B3.5-Higher-Deriv (8 problems) Definition of higher derivatives Velocity, acceleration Derivatives of e x Motion of an object with constant acceleration nth order derivatives of e x, x r, and xe x Motion of an object falling under gravity 3.6 Derivatives of trig functions, B3.6-Trig-Deriv (12 problems) Proof using basic trig limits and addition formula Use of quotient rule to find derivatives of tan cot sec csc nth order derivatives of sin and cos Finding the derivative of tan using quotient rule 3.7 Chain Rule, B3.7-Chain Statement and use of chain rule Using chain rule to differentiate a reciprocal Understand that the generalized power rule, the exponential rule and the shifting and scaling rules are special cases of the chain rule. Statement of chain rule

7 Computing derivatives with chain rule Derivatives of inverse functions, Derivatives of Exponential and Logarithms, B Deriv-Inverse Derivative of inverse functions Derivative of ln(x) Derivative of arcsin(x), arctan(x), and arcsec(x) Students should not know the derivative of b x. They should know to rewrite b x = e x ln(b). Omit material on hyperbolic functions Using formula for derivative of an inverse function Using derivatives as above 3.10 Implicit differentiation, B3.10-Implicit (11 problems) Computing first derivatives by implicit differentiation, omit finding the second derivative by implicit differentiation Revisit derivatives of inverse functions Find tangent line(s) for a curve defined implicitly Use implicit differentiation to find the derivative of an inverse function 3.11 Related rates, B3.11-Rel-Rates (14 problems) Word problems Review formulae for areas and volumes as needed. Students should know the formula for area of plane figures such as triangles, rectangles and circles. Formulae related to three dimensional objects will be provided if needed on exams. Relation between rates of changes of sides and angles in right triangle

8 Problems involving rates of changes for sides of a triangle. Rates of changes for quantities related by volume and area formulae

9 Part C, 10 lectures + 2 Review 4.1 Linear approximation, C4.1-Lin-Approx Definition of the linearization of a function Computing approximations of functions Definition of absolute error and relative error Estimation in applied problems Exam problems Find a linear approximation and compute errors Estimation in a word problem, e.g. volume of paint on a sphere 4.2 Extreme values, C4.2-Extrema (12 problems) Existence of extreme values on a closed interval Failure to attain extreme values on an open interval Fermat s theorem Critical points and finding extreme values on a closed interval Finding extrema Finding critical points Statement of theorem on existence of extrema on closed, bounded intervals Examples to illustrate the failure of the theorem on a closed interval 4.3 The mean value theorem and monotonicity, C4.3-MVT Statement of mean value theorem Estimating functions First derivative test for monotonicity Finding intervals of concavity

10 Finding distance traveled Statement of mean value theorem Use of mean value theorem for estimating functions 4.4 Shape of a graph, C4.4-Monot-Conc 11 problems First derivative test for local extremes Concavity Second derivative test for concavity Second derivative test for local extrema Find intervals of concavity and monotonicity Classify extrema Sketch a graph Match graphs to information about derivatives 2.7 Limits at infinity, C2.7-Limits-Inf (10 problems) Definition Limits of rational functions at infinity Compute limits at infinity 4.5 L Hôpital, C4.5-LHopital Limits of e x /x n Limit of ln(1 + x)/x Limit of (1 + x) 1/x Exam problems Compute limits Recognize when L Hopital does not apply 4.7 Optimization, C4.7-Optimize (15 problems) Formulating a precise optimization problem

11 Criteria for finding extreme values Justifying existence of extrema on an open interval using first derivative (if the first derivative changes sign exactly once, then we have an extrema where the first derivative changes sign). Exam problems Optimization problems. Be able to explain why you have an extrema 4.8 Newton s method, C4.8-Newton (13 problems) Carry out method numerically and graphically Deriving the iteration formula Applications of finding roots: critical points or values of an inverse function Graphical description of Newton s method Find roots Use Newton s method to solve interesting equations. For example to find a critical point or a value of f Anti-derivatives, C4.9-Anti-Deriv (10 problems) Find anti-derivatives Formulation in terms of recovering position from velocity Find anti-derivatives Finding anti-derivatives disguised as recovering position from velocity or recovering velocity from acceleration 5.1 S 5.1 Approximating and computing area, C5.1-Area (15 problems) Definition of midpoint, left and right sums Sigma notation, distributive rule for sums Sums of powers of integers Estimating area with left or right sums (exclude mid-point) Manipulating sums, including using formulae for sums

12 Part D review 5.2 The definite integral, D5.2-Integral Defining area as a limit of Riemann sums The integral as signed area Properties of the integral Applying linearity of the integral Using area to evaluate integrals when the region under the graph is a triangle or part of a circle. 5.3 The fundamental theorem of Calculus, Part I (evaluating integrals) Statement of Fundamental theorem I Evaluating simple integrals. 5.4 The fundamental theorem of Calculus, Part II, D FTC (19 problems) 5.5 The net change theorem (interpretation of FTC), D5.5-Rates (10 points) Exam questions State fundamental theorems I and II Use fundamental theorem to evaluate integrals Differentiating an integral Provide examples to illustrate hypotheses Recovering functions from rates of change: interpretation of the fundamental theorem in applications 5.6 Substitution method, D5.6-Substitution (10 problems) S 5.7, Further transcendental functions D5.7-Transcendental-Fun (10 problems) Exam questions Evaluate definite and indefinite integrals by substitution Be able to change limits in definite integrals Anti-derivatives of 1/x, 1/(1+x 2 ), 1/sqrt(1 x 2 ) and 1/( x x 2 1).

13 5.8 Exponential growth and decay, D5.8-Exp-Gro-Decay (10 problems) Give all solutions to y = ky Formulating simple differential equations Applications of exponential growth and decay including population growth and radioactive decay Exam questions Finding functions of the form f(t) = Ae kt given appropriate information Formulate simple differential equations 6.1 Area of regions in the plane, D6.1-Area (10 problems) Writing areas as an integral Evaluating the integral to find the area Finding areas with roles of x and y interchanged Expressing area between curves as an integral, including problems where the roles of x and y are reversed. Evaluating area

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