Question
What is the slope of the line tangent to the curve y=arctan(4x) at the point at which \(x=\frac{1}{4}\):
A 2
B \(\frac{1}{2}\)
C 0
D \(-\frac{1}{2}\)
E -2
▶️Answer/Explanation
Ans:A
Question
\(\frac{\mathrm{d} }{\mathrm{d} x}(\tan ^{-1}(3x))\)=
A \(3\sec ^{2}(3x)\)
B \(-3\csc ^{2}(3x)\)
C \(\frac{3}{\sqrt{1-(3x)^{2}}}\)
D \(\frac{3}{1+(3x)^{2}}\)
▶️Answer/Explanation
Ans:D
Question
If \(f(x)=\arcsin x\),then \(\lim_{x\rightarrow \frac{1}{2}}\frac{f(x)-f(\frac{1}{2})}{x-\frac{1}{2}}\) is
A 0
B \(\frac{\pi }{6}\)
C \(\frac{\sqrt{3}}{2}\)
D nonexistent
▶️Answer/Explanation
Ans:C
The expression in the limit is the definition of the derivative of arcsin
at \(x=\frac{1}{2}\).The value of the limit is therefore \(\frac{\mathrm{d} }{\mathrm{d} x}\arcsin x|_{x=\frac{1}{2}}=\frac{1}{\sqrt{1-x^{2}}}|_{x=\frac{1}{2}}=\frac{1}{\sqrt{\frac{3}{4}}}=\frac{2}{\sqrt{3}}\)
Question
If which of the following could be the value of a?
A \(\frac{\sqrt{2}}{2}\)
B \(\frac{\sqrt{3}}{2}\)
C \(\sqrt{3}\)
D \(\frac{1}{2}\)
E 2
▶️Answer/Explanation
Ans:B