Find an equation of the tangent line to the curve $y=e^x$ at the point $(1,e)$.

Given the parametric equations $x=t^2-4$ and $y=t^2-2t$, find the equation of the tangent line to the curve at the point where $t=3$.

Find an equation of the tangent plane to the surface $z=x^2+y^2$ at the point $(1,2,5)$.

Find the equation of the tangent line to the curve $y=\sin \left(x\right)+\cos \left(x\right)$ at the point $(0,1)$.

Find an equation of the tangent line to the graph of $f\left(x\right)=x^2+2x$ at the point where $x=1$.

The Eiffel Tower Property Let R be the region between the curves y = e^(-c·x) and y = -e^(-c·x) on the interval [a, ∞), where a ≥ 0 and c > 0.

The center of mass of R is located at (x̄, 0), where x̄ = [∫(a to ∞) x·e^(-c·x) dx] / [∫(a to ∞) e^(-c·x) dx]

(The profile of the Eiffel Tower is modeled by these two exponential curves; see the Guided Project ""The exponential Eiffel Tower"")

b. With a = 0 and c = 2, find the equations of the lines tangent to both curves at x = 0

In Exercises 41–44:

c. Evaluate df/dx at x = a and df⁻¹/dx at x = f(a) to show that

(df⁻¹/dx)|ₓ₌f(a) = 1 / (df/dx)|ₓ₌a

44. f(x) = 2x², x ≥ 0, a = 5

c. Find the slopes of the tangent lines to the graphs of f and g at (1, 1) and (−1, −1) (four tangent lines in all).

c. Find the slopes of the tangent lines to the graphs of h and k at (2, 2) and (−2, −2).