 # Calculus Problems And Their Solutions

## Transcribed Text

1. Suppose 𝑓(𝑥) = (𝑥 + 7)3. Use the definition of the derivative to determine the value 𝑜𝑜 𝑓′ (1), i.e., compute limℎ→0 𝑓(1 + ℎ)— 𝑓(1) ℎ 2. If 𝑓(𝑥) = 3 4𝑥+2 Then 𝑓(𝑥 + ℎ) − 𝑓(𝑥) ℎ = 𝐴 (4𝑥 + 2)(𝐵𝐵 + 𝑑 + 𝐷ℎ) Find 𝐴, 𝐵, 𝐶, and 𝐷. 3. If 𝑓(𝑥) = √9𝑥 + 2 , then 𝑓(𝑥 + ℎ) − 𝑓(𝑥) ℎ = 𝐴 √9𝑥 + 2 + √𝐵𝐵 + 𝐶ℎ + 𝐷 Find 𝐴, 𝐵, 𝐶, 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(𝑥) at the point 𝑥 = 6 to at least three decimal places. In other words determine 𝑓′ (6) where 𝑓(𝑥) = ln(𝑥) HINT: you should compute 𝑓(6 + ℎ) − 𝑓(6) ℎ for small values of ℎ. The natural logarithm ln(𝑥) 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(𝑥+ℎ)−𝑓(𝑥) ℎ = −2ℎ(𝑥+2)−ℎ2 ℎ(𝑥+ℎ+2)2(𝑥+2)2 . Find the slope 𝑚 of the target line at 𝑥 = 4

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