\( F(x)=-\frac{3}{x}+5 \) 1 Determine the equation of the tangent at \( x=3 \)

To find the equation of the tangent line at \( x=3 \), we first need the derivative \( F'(x) \). The derivative of \( F(x)=-\frac{3}{x}+5 \) is \( F'(x)=\frac{3}{x^2} \). Evaluating at \( x=3 \), we get \( F'(3)=\frac{3}{3^2}=\frac{1}{3} \). Now, calculate \( F(3)=-\frac{3}{3}+5=4 \). Using the point-slope form \( y-y_1=m(x-x_1) \) with \( m=\frac{1}{3} \), \( x_1=3 \), and \( y_1=4 \), the tangent line equation is \( y-4=\frac{1}{3}(x-3) \) or \( y=\frac{1}{3}x + 3 \). To visualize tangent lines, think of a rollercoaster track! At any given point, the slope of the tangent line helps us understand how steep the coaster is at that moment. It’s like having a snapshot of the ride—at \( x=3 \), the slope tells us that for every horizontal move of 3 units, the vertical change is just 1 unit! Mathematics has a rich history intertwined with art and architecture. For instance, ancient Greeks used curves to model celestial bodies, laying the groundwork for calculus. Fast forward to the Renaissance, artists like da Vinci utilized perspective—calculating angles and slopes (tangents) to create depth in their works. So, the next time you marvel at a masterpiece, remember, there's a bit of math hidden there!