MTH 151 Midterm: MTH 151 - Term Test 2

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Published on 15 Sep 2018
Department
Course
Professor
Miami University
MTH 151
Term Test 2
Exam Guide
Calculus I
Winter 2018
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Table of Contents
Chapter 3: Differentiation Rules ..................................................................................................... 3
3.1 Polynomial and Exponential Functions ................................................................................. 3
3.2 The Product and Quotient Rules ........................................................................................... 4
3.3 Derivative of Trigonometric Functions.................................................................................. 5
3.4 Chain Rule .............................................................................................................................. 7
3.5 Implicit Differentiation ........................................................................................................ 10
3.6 Derivatives of Logarithmic Functions .................................................................................. 13
3.7 Rates of Change in the Natural and Social Sciences ........................................................... 17
3.8 Exponential Growth and Decay ........................................................................................... 20
3.9 Related Rates ....................................................................................................................... 23
Chapter 4: Applications of Differentiation .................................................................................... 27
4.1 Maximum and Minimum Values ......................................................................................... 27
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Chapter 3: Differentiation Rules
3.1 Polynomial and Exponential Functions
1.

2.

Example:
f(x) = , f’(x) = -2
3.


Example:
f(x) = 3, f’(x) = 3
4.



Example:
f(x) = 



5.
;


Past Exam Question:
1. Find f’(0) if .
Solution: 



2. A point P on the curve  is such that the tangent line at P is parallel to the line
. Find the y-coordinate of the point P.
Solution: 

Since the tangent line is parallel to , then:

When x = 1, y = 1 + 2 4 = -1.
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Document Summary

3. 7 rates of change in the natural and social sciences 17. 3. 1 polynomial and exponential functions: (cid:3031)(cid:3031)(cid:3051) (cid:4666)(cid:1855)(cid:4667)=(cid:882),(cid:1855) (cid:1871) (cid:1855)(cid:1867)(cid:1866)(cid:1871)(cid:1872)(cid:1853)(cid:1866)(cid:1872), (cid:3031)(cid:3031)(cid:3051) (cid:4666)(cid:1876)(cid:3041)(cid:4667)=(cid:1866)(cid:1876)(cid:3041) (cid:2869) f(x) = (cid:1876) (cid:2870), f(cid:859)(cid:894)(cid:454)(cid:895) = -2(cid:1876) (cid:2871, (cid:3031)(cid:3031)(cid:3051)[(cid:1855)(cid:1858)(cid:4666)(cid:1876)(cid:4667)]=(cid:1855)(cid:3031)(cid:3031)(cid:3051)(cid:1858)(cid:4666)(cid:1876)(cid:4667) f(x) = 3(cid:1876)(cid:2872), f(cid:859)(cid:894)(cid:454)(cid:895) = (cid:1007)(cid:1858) (cid:4666)(cid:1876)(cid:2872)(cid:4667)=(cid:885) (cid:886)(cid:1876)(cid:2871)=(cid:883)(cid:884)(cid:1876)(cid:2871, (cid:3031)(cid:3031)(cid:3051)[(cid:1858)(cid:4666)(cid:1876)(cid:4667) (cid:1859)(cid:4666)(cid:1876)(cid:4667)]= (cid:3031)(cid:3031)(cid:3051)(cid:1858)(cid:4666)(cid:1876)(cid:4667) (cid:3031)(cid:3031)(cid:3051)(cid:1859)(cid:4666)(cid:1876)(cid:4667) f(x) = (cid:1876)(cid:2872) (cid:888)(cid:1876)(cid:2870)+(cid:886)= (cid:3031)(cid:3031)(cid:3051)(cid:1876)(cid:2872) (cid:3031)(cid:3031)(cid:3051)(cid:888)(cid:1876)(cid:2870)+ (cid:3031)(cid:3031)(cid:3051)(cid:886)=(cid:886)(cid:1876)(cid:2871) (cid:883)(cid:884)(cid:1876, (cid:3031)(cid:3031)(cid:3051)(cid:4666)(cid:1857)(cid:3051)(cid:4667)=(cid:1857)(cid:3051); (cid:888). (cid:1856)(cid:1856)(cid:1876)(cid:4666)(cid:1853)(cid:3051)(cid:4667)=(cid:1853)(cid:3051)(cid:1864)(cid:1866)(cid:1853) (cid:1005). =n(cid:885)(cid:4666)(cid:885)(cid:3051) (cid:885) (cid:3051)(cid:4667) f (cid:4666)(cid:882)(cid:4667)=n(cid:885)(cid:4666)(cid:883) (cid:883)(cid:4667)=(cid:882: a point p on the curve (cid:1825)= (cid:1824)(cid:2779)+(cid:2779)(cid:2206) (cid:2781) is such that the tangent line at p is parallel to the line (cid:1825)= (cid:2781)(cid:1824)+(cid:2778)(cid:2778). Find the y-coordinate of the point p. y= (cid:2870)+(cid:884)(cid:1876) (cid:886) y =(cid:884)+(cid:884) Since the tangent line is parallel to y=(cid:886)+(cid:883)(cid:883), then: (cid:884)+(cid:884)=(cid:886),=(cid:883). The product rule (cid:2188)(cid:4666)(cid:2206)(cid:4667)= (cid:2206)(cid:2189)(cid:4666)(cid:2206)(cid:4667), g(cid:894)(cid:1008)(cid:895) = (cid:1006), g"(cid:894)(cid:1008)(cid:895) = (cid:1007),fi(cid:374)d f"(cid:894)(cid:1008)(cid:895). The quotient rule f(x) = (cid:2187)(cid:2206) (cid:2778)(cid:2206)+(cid:2778), fi(cid:374)d f"(cid:894)(cid:454)(cid:895). Past exam questions: f(x) is differe(cid:374)tia(cid:271)le. f(cid:894)(cid:1005)(cid:895)=(cid:1006),f(cid:894)(cid:1006)(cid:895)=(cid:1007),f"(cid:894)(cid:1005)(cid:895)=(cid:1008), f"(cid:894)(cid:1006)(cid:895)=(cid:1009), f"(cid:894)(cid:1007)(cid:895)=(cid:1010). If g(cid:894)(cid:454)(cid:895) = f[ xf[ xf(x) ] ], fi(cid:374)d g"(cid:894)(cid:1005)(cid:895).

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