Calculus II Test #1A Solutions

I. Evaluate the given limits

1. a) MATH

b) MATH

2. a) MATH $\ $write as MATH so

ln(MATH

=MATH Therefore $L=e^{0}=1$

b) MATH

II. Find the derivative dy/dx for the following functions.

3. MATH

MATH

4. $y=\ln (\cosh (x))$

MATH

5. MATH => $y^{4}=y+\ln (x)$

MATH => MATH => MATH

6. $y=e^{\tan (x^{2})}$

MATH

7. MATH

MATH

8. MATH

MATH

9. MATH write using MATH

MATH

MATH

10. MATH

MATH

11 . MATH

MATH

12. MATH

MATH


III. Evaluate the following integrals.

13. MATH

Omitted because I made a mistake on the problem

14. MATH

Let MATH

15. MATH

Let MATH

16. MATH

Let MATH

17. MATH

Let MATH

IV. Other problems:

18. Derive the formula for the derivative of $\cosh ^{-1}(x)$ by using the inverse function method of differentiation.

$y=\cosh ^{-1}(x)$ => $x=\cosh (y)$ => MATH

MATH

19. Find $\frac{dy}{dx}$ for MATH where x occurs in the power an infinite number of times.

Write this as $y=x^{y},$ then $\ln (y)=y\ln x$ => MATH

or multiplying both sides by $xy$: MATH => MATH

MATH

20. Find the equation of the tangent line to the curve $y=(\ln (x)+2)^{2}$ at $\ x=1$

At $x=1$, MATHAt $x=1,$

MATH => $y=4x+b$

$4=4(1)+b\ =>b=0:$ $y=4x$