C cole123 New member Joined Mar 11, 2010 Messages 1 Mar 11, 2010 #1 find the function f(x) the graph passes through (1,2) and whose tangent line at the point (x,F(x)) has a slope of 1/x^3
find the function f(x) the graph passes through (1,2) and whose tangent line at the point (x,F(x)) has a slope of 1/x^3
S soroban Elite Member Joined Jan 28, 2005 Messages 5,584 Mar 11, 2010 #2 Hello, cole123! \(\displaystyle \text{Find the function }f(x)\,\text{ if the graph passes through (1,2)}\) \(\displaystyle \text{and whose tangent line at the point }(x,f(x))\text{ has a slope of }\frac{1}{x^3}\) Click to expand... \(\displaystyle \text{We are given: }\;f'(x) \:=\:x^{-3}\) . . \(\displaystyle \text{Hence: }\:f(x) \;=\;-\frac{1}{2}x^{-2} + C \;=\;-\frac{1}{2x^2} + C\) \(\displaystyle \text{We are given: }\;f(1) \,=\,2 \quad\Rightarrow\quad -\frac{1}{2(1^2)} + C \:=\:2 \quad\Rightarrow\quad C \:=\:\frac{5}{2}\) \(\displaystyle \text{Therefore: }\;\boxed{f(x) \;=\;-\frac{1}{2x^2} + \frac{5}{2}}\)
Hello, cole123! \(\displaystyle \text{Find the function }f(x)\,\text{ if the graph passes through (1,2)}\) \(\displaystyle \text{and whose tangent line at the point }(x,f(x))\text{ has a slope of }\frac{1}{x^3}\) Click to expand... \(\displaystyle \text{We are given: }\;f'(x) \:=\:x^{-3}\) . . \(\displaystyle \text{Hence: }\:f(x) \;=\;-\frac{1}{2}x^{-2} + C \;=\;-\frac{1}{2x^2} + C\) \(\displaystyle \text{We are given: }\;f(1) \,=\,2 \quad\Rightarrow\quad -\frac{1}{2(1^2)} + C \:=\:2 \quad\Rightarrow\quad C \:=\:\frac{5}{2}\) \(\displaystyle \text{Therefore: }\;\boxed{f(x) \;=\;-\frac{1}{2x^2} + \frac{5}{2}}\)
