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Functions operations and Inverse

100

Given 𝑓(𝑥) = 𝑥² + 6 and 𝑔(𝑥) = 𝑥 − 1 

Find (𝑓 + 𝑔)(𝑥). 

 (𝑓+𝑔)(𝑥)=(𝑥²+6)+(𝑥−1)=𝑥²+𝑥+5  

200

Given 𝑓(𝑥) = 𝑥² + 6 and 𝑔(𝑥) = 𝑥 − 1 

 Find (𝑓∙𝑔)(𝑥). 

 (𝑓∙𝑔)(𝑥)=(𝑥²+6 )(𝑥−1)=𝑥³−𝑥²+6𝑥−6  

300

 If 𝑓(𝑥)=2𝑥+5 and 𝑔(𝑥)=3𝑥 , find 𝑔(𝑓(1)). 

 𝑓(1)=2(1)+5=7 

𝑔(𝑓(1))=𝑔(7)=3(7)=21  

400

Find the inverse of the function 𝑓(𝑥)=√𝑥−1 . 

𝑓⁻¹(𝑥) = 𝑥²+1  

500

If 𝑔(𝑥)=𝑥−4/𝑥−5 , find the range of 𝑔⁻¹ . 

Range = (−∞,5)∪(5,∞) 

Calculus:Derivatives and Integrals

100

Find the derivative of the function 𝑓(𝑥)=4𝑥³+3𝑥²+5𝑥+2. 

𝑓′(𝑥)=12𝑥²+6𝑥+5  

200

If 𝑓(𝑥)=2𝑥³−4𝑥²+5. Find 𝑓′′(2). 

𝑓′(𝑥)=6𝑥²−8𝑥 

𝑓′′(𝑥)=12𝑥−8 

𝑓′′(2)=12(2)−8=16  

300

Find the slope of the tangent line to the graph of 𝑓(𝑥)=3𝑥²−4𝑥+6 at 𝑥=3 ? 

𝑓′(𝑥)=6𝑥−4 

𝑚=6(3)−4=14  

400

Find ∫(2𝑥²+5𝑥+7)𝑑𝑥 

2𝑥³/3 + 5𝑥²/2 + 7𝑥 + 𝑐  

500

The velocity of a flea’s jump can be defined as 𝑣(𝑡)=−10𝑡+4, where 𝑡 is given in seconds and velocity is given in meters per second. Find the position function 𝑠(𝑡) for the flea’s jump. Assume that for 𝑡= 0,𝑠(𝑡) = 5. 

𝑠(𝑡)=−5𝑡²+4𝑡+5  

Polar Coordinates

100

Which of the following represent another pair of polar coordinates for point A (3,225°)? 

A. (−3,135°) 

B. (3,135°) 

C. (3,45°)

D. (−3,45°)

D. (−3,45°)

200

Find the distance P1P2 between each pair of points to the nearest tenth degree. 

P1(3,40°),P2 (5,130°) 

P1P2=√34=5.8 

300

Find the rectangular coordinates of the point with the given polar coordinates. 𝑃(−2,𝜋)  

The rectangular coordinates of P are (2,0). 

400

Convert the polar equation to rectangular form. 𝑟=3𝑠𝑒𝑐𝜃  

𝑥=3 

500

For the complex number given below find the absolute value. 

𝑧=√2+√2 𝑖  

|𝑧|=√(√2)²+(√2)²=2 

Sequences and Series

100

Find the 16th term of the arithmetic sequence −18,−6,6,18,… 

𝑎_𝑛=𝑎1+(𝑛−1)𝑑 

𝑎₁₆=−18+(16−1)(12)

 𝑎₁₆=162  

200

Find the sum of the geometric series Σ4(−3)^𝑘−1. from 1 to 7

𝑆₇ = 2188  

300

In a physics experiment, a steel ball on a flat track is accelerated, and then allowed to roll freely. After the first minute, the ball rolled 120 meters. Each minute the ball travels only 40% as far as it did during the preceding minute. How far does the ball travel? 

𝑆= 𝑎₁/1−𝑟 = 120/1−0.4 = 200 m

400

Determine whether each geometric series is convergent or divergent. 

21+63+189+⋯  

divergent as r = 3 >1

500

A pendulum travels 12 centimeters on its first swing and 95% of that distance on each swing thereafter. Find the total distance traveled by the pendulum when it comes to rest. 

𝑆= 𝑎₁/1−𝑟 = 12/1−0.95 = 240 cm