NAME ________________________________ DATE _________________ Part 1 Derivatives with Types power, exp, and ln Circle your answers 1. Find f ‘ ( x) : 𝑓(𝑥) = 𝑙𝑛( 1 − 2𝑥 3 ) Type _______ U= _______ U’ = _______ 2. Find f ‘ ( x) : 3. Find dy/dx: 𝑓(𝑥) = 3𝑒 −3𝑥 Type _______ U= _______ U’ = _______ 𝑓(𝑥) = 4𝑥 𝑙𝑛( 𝑥) {Hint: uses product rule} 4. 𝑑 𝑑𝑥 [3 𝑙𝑛( 1 𝑒 2𝑥 )] Type _______ U= _______ U’ = _______ 5. Use the Quotient rule to find 𝑓′(𝑥) for 𝑓(𝑥) = 2𝑥 2 𝑒 2𝑥 Part 2 Integration using types power, exp, and ln Circle your answers please. Types power, exp and ln 6. ∫(4𝑥 3 − 2𝑥 1)𝑑𝑥 Type _______ U= _______ U’ = _______ 3 3 7. ∫( √𝑥 𝑒 3𝑥 )𝑑𝑥 Types ____ ____ _____ U= _______ U’ = _______ 𝑥 Show insertion and compensation for all integration except for U = x 1 8. ∫0 1 𝑥 3 1 𝑥 2 𝑑𝑥 {rounded to 3 decimal places} Type _______ U= _______ U’ = _______ 9. ∫(𝑥 3 2)8 2𝑥 2 𝑑𝑥 Type _______ U= _______ U’ = _______ Show insertion and compensation for all integration 1 10. Evaluate: ∫0 6𝑒 (2𝑡−1) 𝑑𝑡 {rounded to 2 decimal places} `Type _______ U= _______ U’ = _______ Show insertion and compensation 11. ∫ 6𝑥 (𝑥 2 1)4 𝑑𝑥 Type _______ U= _______ U’ = _______ Show insertion and compensation 12. ∫ 𝑥√2𝑥 2 1 𝑑𝑥 Type _______ U= _______ U’ = _______ Show insertion and compensation 13. Shade and find the area of the region between 𝑦 = 𝑥 2 − 1 and 𝑦 = 2𝑥 2 1 from x = 0 to x = 2. Sketch both curves and shade the area that we are finding. Set up the integral and evaluate. 14. Given that 𝒇′′ (𝒙) = 𝒙 − 𝟐 find 𝒇(𝒙) given that f(0) = 1 and f’(1) = 2 You will need to find two constants. Bonus A company’s marginal cost is given by: 1 𝐶′(𝑥) = = rewrite = __________ Fixed costs are $50 (That is: C(0) = √2𝑥 1 $50) Rewrite C’(x) _____________________ A. Use an integral to find the cost function C(x)in radical form. Type _______ U= _______ U’ = _______ B. Find the cost of the first 10 items (to 2 decimal places)