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# Calculus I ## Review Problems for Exam I ### Chapter 1 1. (1.1) For the function $f$ whose graph is given, estimate the value of $f(2)$. *The value of $f(2)$ is approximately 1.4* 2. (1.1) Find the domain an range of the function $f$ whose graph is given. *The domain of the function...

# Calculus I ## Review Problems for Exam I ### Chapter 1 1. (1.1) For the function $f$ whose graph is given, estimate the value of $f(2)$. *The value of $f(2)$ is approximately 1.4* 2. (1.1) Find the domain an range of the function $f$ whose graph is given. *The domain of the function is [-2, 2] and the range is [-1, 2]* 3. (1.1) Determine whether the curve is the graph of a function of $x$. If it is, state the domain and range of the function. *The curve is the graph of a function of x. The domain is [-3, 3] and the range is [-2, 3]* 4. (1.1) Sketch a rough graph of the function without using a calculator. $y = x^3$ $y = (x-1)^3$ $y = -x^3$ $y = x^3 + 2$ 5. (1.2) Find an expression for the function whose graph is the line segment joining the points $(1, -3)$ and $(5, 7)$. *The function is $f(x) = \frac{5}{2}x - \frac{11}{2}$$ 6. (1.2) Given that $f(x) = \sqrt{x}$ and $g(x) = x-1$, find $f \circ g, g \circ f, f \circ f, g \circ g$ and their domains. *$(f \circ g)(x) = \sqrt{x - 1}$, Domain: $[1, \infty)$ $(g \circ f)(x) = \sqrt{x} - 1$, Domain: $[0, \infty)$ $(f \circ f)(x) = \sqrt{\sqrt{x}}$, Domain: $[0, \infty)$ $(g \circ g)(x) = x - 2$, Domain: $(-\infty, \infty)$* 7. (1.3) Find a formula for the inverse of the function $f(x) = \sqrt{10 - 3x}$. *The inverse function is $f^{-1}(x) = -\frac{1}{3}(x^2 - 10)$* 8. (1.5) Determine the infinite limit. $\lim_{x \rightarrow 5^{+}} \frac{6}{x - 5}$ $\lim_{x \rightarrow 5^{-}} \frac{6}{x - 5}$ $\lim_{x \rightarrow 1} \frac{2 - x}{(x - 1)^2}$ $\lim_{x \rightarrow -2^{+}} \frac{x - 1}{x(x + 2)}$ 9. (1.6) Find the limit. $\lim_{x \rightarrow \infty} \frac{1}{x^2}$ $\lim_{x \rightarrow -\infty} \frac{1}{x^2}$ $\lim_{x \rightarrow \infty} \frac{3x + 5}{x - 4}$ $\lim_{x \rightarrow \infty} \frac{x^2 + 1}{x^4 - 3}$ $\lim_{x \rightarrow -\infty} \frac{x^2 + x}{3 - x}$ $\lim_{x \rightarrow \infty} (\sqrt{x^2 + ax} - x)$ 10. (1.7) Use continuity to evaluate the limit. $\lim_{x \rightarrow 4} (5 + \sqrt{x})$ $\lim_{x \rightarrow \pi} \sin(x + \sin x)$ 11. (1.7) Find the value(s) of $c$ that makes $f$ continuous everywhere. $f(x) = \begin{cases} x^2 & \text{if } x < c \\ x + 1 & \text{if } x \geq c \end{cases}$ 12. (1.8) Find the limit. $\lim_{x \rightarrow 0} \frac{\sin 3x}{x}$ $\lim_{t \rightarrow 0} \frac{\tan 6t}{\sin 2t}$ $\lim_{x \rightarrow 0} \frac{\sin x}{5x^2 + x}$ $\lim_{\theta \rightarrow 0} \frac{\sin \theta}{\theta + \tan \theta}$ ### Chapter 2 1. (2.1) Find an equation of the tangent line to the curve $y = \frac{2}{x}$ at the point $(2, 1)$. *The equation of the tangent line is $y = -\frac{1}{2}x + 2$* 2. (2.2) If a ball is thrown into the air with a velocity of 40 ft/s, its height in feet $t$ seconds later is given by $y = 40t - 16t^2$. Find the velocity when $t = 2$. *The velocity when $t = 2$ is -24 ft/s* 3. (2.3) Find the derivative of the function. $f(x) = 1.4x^2 - 0.6x + 1$ $g(x) = \sqrt{x} + \frac{1}{\sqrt{x}}$ $y = x^2 + \frac{2}{x}$ $y = \frac{x^2 + 4x + 3}{\sqrt{x}}$ 4. (2.4) Find the derivative of the function. $y = e^x \cos x$ $y = \frac{x^2}{e^x}$ $f(t) = \frac{t^2 - 1}{t^2 + 1}$ $f(x) = \frac{ax + b}{cx + d}$ 5. (2.5) Find $y'$ and $y''$. $y = \sqrt{1 - x^2}$ $y = e^{-5x}$ 6. (2.6) Find the derivative of the function. $y = \sin(x^2)$ $y = \tan(\sqrt{x})$ $y = \sin^2(x)$ $y = \sqrt{\sin x}$ $y = \sin(\sqrt{x})$ $y = e^{\sin 2\theta}$ $y = \sin(e^{\sqrt{x}})$ 7. (2.7) Find $dy/dx$ by implicit differentiation. $x^2 - xy + y^2 = 5$ $x^3 + x^2y + 4y^2 = 6$