1. The problem asks for the third and fourth terms in the Taylor series expansion of a function $f(x)$ about the point $a$. 2. The Taylor series formula for a function $f(x)$ expanded about $a$ is: $$f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f^{(3)}(a)}{3!}(x-a)^3 + \cdots$$ 3. The first term is $f(a)$, the second term is $f'(a)(x-a)$, which are given. 4. The third term is the term with the second derivative: $$\frac{f''(a)}{2!}(x-a)^2 = \frac{f''(a)}{2}(x-a)^2$$ 5. The fourth term is the term with the third derivative: $$\frac{f^{(3)}(a)}{3!}(x-a)^3 = \frac{f^{(3)}(a)}{6}(x-a)^3$$ 6. So, the third and fourth terms are: $$\frac{f''(a)}{2}(x-a)^2 + \frac{f^{(3)}(a)}{6}(x-a)^3$$ 7. These terms represent the curvature and the rate of change of curvature of the function at $a$, respectively, and are important for approximating $f(x)$ near $a$ more accurately.