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1. Suppose 𝑓(π‘₯) = (π‘₯ + 7)3. Use the definition of the derivative to determine the value π‘œπ‘œ 𝑓′ (1), i.e., compute limβ„Žβ†’0 𝑓(1 + β„Ž)β€” 𝑓(1) β„Ž 2. If 𝑓(π‘₯) = 3 4π‘₯+2 Then 𝑓(π‘₯ + β„Ž) βˆ’ 𝑓(π‘₯) β„Ž = 𝐴 (4π‘₯ + 2)(𝐡𝐡 + 𝑑 + π·β„Ž) Find 𝐴, 𝐡, 𝐢, and 𝐷. 3. If 𝑓(π‘₯) = √9π‘₯ + 2 , then 𝑓(π‘₯ + β„Ž) βˆ’ 𝑓(π‘₯) β„Ž = 𝐴 √9π‘₯ + 2 + √𝐡𝐡 + πΆβ„Ž + 𝐷 Find 𝐴, 𝐡, 𝐢, and 𝐷. 4. You will need a calculator with the natural logarithm function for this problem. Use the definition of the derivative to guess the value of the derivative of 𝑓(π‘₯) at the point π‘₯ = 6 to at least three decimal places. In other words determine 𝑓′ (6) where 𝑓(π‘₯) = ln(π‘₯) HINT: you should compute 𝑓(6 + β„Ž) βˆ’ 𝑓(6) β„Ž for small values of β„Ž. The natural logarithm ln(π‘₯) is found on most calculators. Be sure you do not use the base 10 logarithm which is usually just written as log(x). 5. Suppose that 𝑓(π‘₯+β„Ž)βˆ’π‘“(π‘₯) β„Ž = βˆ’2β„Ž(π‘₯+2)βˆ’β„Ž2 β„Ž(π‘₯+β„Ž+2)2(π‘₯+2)2 . Find the slope π‘š of the target line at π‘₯ = 4

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Calculus Problems And Their Solutions
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