这是份Brock University的Math 1P97 calculus with Applications exam辅导案例，学生在我们的帮助下获得了满意的成绩。
- The domain of $f(x)=\ln (x+1)$ can be written:
a. $(-\infty,-1) \cup(1, \infty)$
b. $[-1, \infty)$
C. $(-1, \infty)$
d. $(0, \infty)$
e. none of the above - If $f(x)=\sqrt{x+1}$ and $g(x)=e^{2 x}$ then $f \circ g$ is
a. $e^{2 x+1}$
b. Undefined
c. $\sqrt{e^{2 x}+1}$
d. $e^{2 \sqrt{x+1}}$
e. $e^{\sqrt{2 x+1}}$ - Find $\lim _{x \rightarrow-2} \frac{x+2}{x^{2}-4}$
a. Limit does not exist
b. 0
c. $-1 / 4$
d. $-4$
e. Infinity - Find $\lim _{x \rightarrow \infty} \frac{x^{3}+2 x+2}{x^{3}-4 x}$
a. $\infty$
b. $-\infty$
C. 1
d. $-1$
e. Limit does not exist
- Let $f$ be the function defined below. Which of the statements about $f$ are true?
$$
f(x)= \begin{cases}\frac{x^{3}-x^{2}}{x-1}, & \text { for } x \neq 1 \ 3 x & \text { for } x=1\end{cases}
$$
Statement I. $f$ has a limit at $x=1$.
Statement II. $f$ is continuous at $x=1$.
Statement III. $f$ is differentiable at $x=1$.
a. III only
b. II and III only
c. I and II only
d. All statements are true
e. I only - Consider the function $f(x)=5 x \ln \left(x^{3}-4\right)$. Find $f^{\prime}(2)$
a. $36.9$
b. $13.9$
c. $9.43$
d. $66.9$
e. none of the above - Find the slope of the tangent line to $x y^{2}+e^{y}=x$ at $(1,0)$.
a. 3
b. 2
c. 1
d. 0
e. None of the above - Which formula represents the limit definition of the derivative of the function $f(x)$ ?
a. $f^{\prime}(x)=\lim {h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ b. $f^{\prime}(x)=\lim {h \rightarrow 0} \frac{f(x+h)}{h}$
c. $f^{\prime}(x)=\lim {h \rightarrow 0} \frac{f(x)}{h}$ d. $f^{\prime}(x)=\lim {x \rightarrow 0} f(x)$
e. $f^{\prime}(x)=\lim _{x \rightarrow 0} \frac{f(x+h)-f(x)}{h}$
For the next two questions use the following equation for the position of an object: $s(t)=6 t^{2}+t+8$ where $t$ is measured in seconds and $s$ is measured in meters.
- What is the position of the object at time $t=10$ seconds?
a. $618 \mathrm{~m}$
b. $121 \mathrm{~m}$
c. $309 \mathrm{~m}$
d. $60.5 \mathrm{~m}$
e. $8 \mathrm{~m}$ - What is the velocity at time $t=10$ seconds?
a. $121 \mathrm{~m} / \mathrm{sec}$
b. $618 \mathrm{~m} / \mathrm{sec}$
c. $60.5 \mathrm{~m} / \mathrm{sec}$
d. $309 \mathrm{~m} / \mathrm{sec}$
e. None of the above - Let $y=x^{3}+3 x^{2}+e^{x}$. Find the fourth derivative of $y$.
a. 0
b. 1
c. $x e^{x-1}$
d. $e^{x-1}$
e. $e^{x}$ - Let $y=\frac{x^{2}+1}{\ln x}$. Find $y^{\prime}$.
a. $\frac{(2 x) \ln (x)-(1 / x)\left(x^{2}+1\right)}{\ln (x)}$
d. $\frac{(2 x) \ln (x)+(1 / x)\left(x^{2}+1\right)}{(\ln (x))^{2}}$
b. $\frac{(2 x) \ln (x)-(1 / x)\left(x^{2}+1\right)}{(\ln (x))^{2}}$
e. $(2 x) \ln (x)-(1 / x)\left(x^{2}+1\right)$
c. None of these answers
MATH 1P97
Calculus With Applications
Lines, polynomials, logarithms and exponential functions; two-sided limits; rates of change using derivatives; max and min of functions using derivatives; higher derivatives and concavity; area under a curve using integrals; optimization of functions of two variables using partial derivatives; growth and decay using differential equations; applications to many different disciplines; use of computer algebra systems.
Lectures, 3 hours per week.
Restriction: not open to Mathematics (single or combined) majors.
Prerequisite(s): MATH 1P20 or one grade 12 mathematics credit.
Note: designed for students in Biological Sciences, Biotechnology, Business, Earth Sciences, Economics, Environmental Geoscience, Geography and Medical Sciences. Not open to students with credit in any university calculus course. Major credit will not be granted to Mathematics majors. This course may be offered in multiple modes of delivery. The method of delivery will be listed on the academic timetable, in the applicable term.
