### Home > CALC > Chapter 6 > Lesson 6.3.1 > Problem6-87

6-87.

Consider the function $f(x) = −3^x$ .

1. Find the equation of the tangent line to $f(x) = −3^x$ at $x = 0$.

Evaluate $f^\prime(0)$ to find the slope, and evaluate $f(0)$ to find the $y$-intercept. Use the point-slope formula to create the equation of the tangent line.

2. Approximate $x = 0.1$ using part (a).

Use the tangent line at $x = 0$ to evaluate $x = 0.1$. This will approximate $f(0.1)$. Why approximate?
Since a tangent is a linear equation, it is easier to evaluate than $f(x)$.

3. Use $f^{\prime\prime}(x)$ to justify whether part (b) is an under or over approximation of the actual value.

A concave up graph has tangent lines that are below the curve. Sketch it, you will see.
And a concave down graph has tangent lines that are above the curve.

Overestimate since $f^{\prime\prime}(0) = −(\operatorname{ln}3)^2 · 3^0 = −(\operatorname{ln}3)^2 < 0$