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APBC:9914011Let Evaluate the following limits, allowing the symbols \infty and -\infty: Identify every vertical asymptote of y=…

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Limits ContinuityAP Calculus BC Problem Solving

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APBC:9914012Let Evaluate \lim_{x\to\infty}R(x) and \lim_{x\to-\infty}R(x).

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APBC:9914013Evaluate First identify a left-hand tail of the real line on which the expression is defined as a real number.

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APBC:9914014(a) Evaluate using the substitution u=5/x and the standard limit \lim_{u\to 0}(\sin u)/u=1.

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Limits ContinuityAP Calculus BC Problem Solving

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APBC:9914015Define Evaluate f(2), \lim_{x\to 2^-}f(x), and \lim_{x\to 2^+}f(x).

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Limits ContinuityAP Calculus BC Problem Solving

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APBC:9914016For a real parameter c, define Find the unique value of c for which f_c is continuous at x=4.

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APBC:9914017Define Find the exact real domain of g.

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APBC:9914018Let p(x)=x^3+4x-6.

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APBC:9914019Let (a) Evaluate \lim_{x\to 0}f(x).

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APBC:9914020A continuous function T gives the outdoor temperature in degrees Fahrenheit t hours after 6:00 a.m., for 0\le t\le 9.

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APBC:9914021Let f(x)=\dfrac{1}{3x-1}.

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APBC:9914022For real parameters a and b, define (a) Determine all pairs (a,b) for which f is continuous at x=3.

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APBC:9914023Let h(x)=\lvert x^2-4\rvert.

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APBC:9914024Starting from the limit definition of the derivative, prove that You may use the cosine angle-addition formula and…

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APBC:9914025Let f(x)=x^{4/3}, defined for every real x by f(x)=\bigl(\sqrt[3]{x}\bigr)^4.

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APBC:9914026Let (a) Find f'(x).

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APBC:9914027Let (a) Find F'(x), identifying explicitly where the product rule is used and where the chain rule is used.

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APBC:9914028Starting from \tan x=\dfrac{\sin x}{\cos x} and the known derivatives of sine and cosine, use the quotient rule to…

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APBC:9914029Let (a) Find F'(x), showing both the product rule and the chain-rule factors on the exponential and on the cosine.

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APBC:9914030Differentiable functions f and g take the exact values Define (a) Compute H(2).

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APBC:9914031The curve x^2+xy+y^2=7 in the xy-plane contains the point (2,1).

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Differentiation Definition RulesAP Calculus BC Problem Solving

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APBC:9914032The curve 9x^2+4y^2=36 in the xy-plane is closed and bounded.

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APBC:9914033The curve x^3+y^3=6xy contains the point (3,3).

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APBC:9914034The curve e^{xy}+x+2y=e+3 contains the point (1,1).

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APBC:9914035On the first quadrant x>0, y>0, the curve x^{y}=y^{x} contains both (2,4) and (4,2).

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APBC:9914036The circle x^2+y^2=13 contains the point (2,3).

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APBC:9914037A differentiable one-to-one function f is defined on an open interval containing x=2 and satisfies f(2)=5 and f'(2)=4.

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APBC:9914038Let f(x)=\arctan(2x), where \arctan denotes the inverse tangent with range \bigl(-\frac{\pi}{2},\frac{\pi}{2}\bigr).

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APBC:9914039Let f be twice differentiable and one-to-one on an open interval containing x=1, with Let g=f^{-1}, and assume g is…

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APBC:9914040Let f(x)=x^5+x.

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APBC:9914041For x>0, let y=x^{\cos x}.

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APBC:9914042A 20-foot ladder leans against a vertical wall, with its foot on horizontal ground.

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APBC:9914043An inverted conical tank (vertex down) is 10 feet deep and has a circular top of radius 4 feet.

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APBC:9914044A spherical balloon remains spherical as it expands.

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APBC:9914045A circular oil slick remains circular while its radius increases at the constant rate of 0.5 inches per second.

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APBC:9914046A lamppost is 18 feet tall.

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APBC:9914047An airplane flies due east at a constant altitude of 400 meters.

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APBC:9914048A radar station at the origin of the coordinate plane tracks a target whose position at time t minutes is \bigl(x(t…

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APBC:9914049Let f(x)=x^3-12x.

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APBC:9914050Let f(x)=x^3-3x^2+1 on the closed interval [0,4].

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APBC:9914011Let Evaluate the following limits, allowing the symbols \infty and -\infty: Identify every vertical asymptote of y=…

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APBC:9914012Let Evaluate \lim_{x\to\infty}R(x) and \lim_{x\to-\infty}R(x).

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APBC:9914013Evaluate First identify a left-hand tail of the real line on which the expression is defined as a real number.

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APBC:9914014(a) Evaluate using the substitution u=5/x and the standard limit \lim_{u\to 0}(\sin u)/u=1.

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APBC:9914015Define Evaluate f(2), \lim_{x\to 2^-}f(x), and \lim_{x\to 2^+}f(x).

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APBC:9914016For a real parameter c, define Find the unique value of c for which f_c is continuous at x=4.

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APBC:9914017Define Find the exact real domain of g.

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APBC:9914018Let p(x)=x^3+4x-6.

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APBC:9914019Let (a) Evaluate \lim_{x\to 0}f(x).

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APBC:9914020A continuous function T gives the outdoor temperature in degrees Fahrenheit t hours after 6:00 a.m., for 0\le t\le 9.

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APBC:9914021Let f(x)=\dfrac{1}{3x-1}.

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APBC:9914022For real parameters a and b, define (a) Determine all pairs (a,b) for which f is continuous at x=3.

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APBC:9914023Let h(x)=\lvert x^2-4\rvert.

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APBC:9914024Starting from the limit definition of the derivative, prove that You may use the cosine angle-addition formula and…

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APBC:9914025Let f(x)=x^{4/3}, defined for every real x by f(x)=\bigl(\sqrt[3]{x}\bigr)^4.

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APBC:9914026Let (a) Find f'(x).

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APBC:9914027Let (a) Find F'(x), identifying explicitly where the product rule is used and where the chain rule is used.

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APBC:9914028Starting from \tan x=\dfrac{\sin x}{\cos x} and the known derivatives of sine and cosine, use the quotient rule to…

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APBC:9914029Let (a) Find F'(x), showing both the product rule and the chain-rule factors on the exponential and on the cosine.

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APBC:9914030Differentiable functions f and g take the exact values Define (a) Compute H(2).

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APBC:9914031The curve x^2+xy+y^2=7 in the xy-plane contains the point (2,1).

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Differentiation Definition RulesAP Calculus BC Problem Solving

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APBC:9914032The curve 9x^2+4y^2=36 in the xy-plane is closed and bounded.

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APBC:9914033The curve x^3+y^3=6xy contains the point (3,3).

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APBC:9914034The curve e^{xy}+x+2y=e+3 contains the point (1,1).

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APBC:9914035On the first quadrant x>0, y>0, the curve x^{y}=y^{x} contains both (2,4) and (4,2).

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APBC:9914036The circle x^2+y^2=13 contains the point (2,3).

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APBC:9914037A differentiable one-to-one function f is defined on an open interval containing x=2 and satisfies f(2)=5 and f'(2)=4.

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APBC:9914038Let f(x)=\arctan(2x), where \arctan denotes the inverse tangent with range \bigl(-\frac{\pi}{2},\frac{\pi}{2}\bigr).

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Differentiation Definition RulesAP Calculus BC Problem Solving

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APBC:9914039Let f be twice differentiable and one-to-one on an open interval containing x=1, with Let g=f^{-1}, and assume g is…

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Differentiation Definition RulesAP Calculus BC Problem Solving

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APBC:9914040Let f(x)=x^5+x.

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APBC:9914041For x>0, let y=x^{\cos x}.

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Differentiation Definition RulesAP Calculus BC Problem Solving

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APBC:9914042A 20-foot ladder leans against a vertical wall, with its foot on horizontal ground.

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Differentiation Definition RulesAP Calculus BC Problem Solving

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APBC:9914043An inverted conical tank (vertex down) is 10 feet deep and has a circular top of radius 4 feet.

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APBC:9914044A spherical balloon remains spherical as it expands.

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APBC:9914045A circular oil slick remains circular while its radius increases at the constant rate of 0.5 inches per second.

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APBC:9914046A lamppost is 18 feet tall.

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APBC:9914047An airplane flies due east at a constant altitude of 400 meters.

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APBC:9914048A radar station at the origin of the coordinate plane tracks a target whose position at time t minutes is \bigl(x(t…

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Differentiation Definition RulesAP Calculus BC Problem Solving

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APBC:9914049Let f(x)=x^3-12x.

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Differentiation Definition RulesAP Calculus BC Problem Solving

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APBC:9914050Let f(x)=x^3-3x^2+1 on the closed interval [0,4].

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Differentiation Definition RulesAP Calculus BC Problem Solving

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APBC:9914001On the interval [-2,6], the graph of a function r consists of three line segments: the segment joining (-2,5) to (0…

APBC:9914002Selected values of a function g are given in the table below.

APBC:9914003Suppose Using only limit laws, evaluate

APBC:9914004Evaluate Show the factoring that is valid for x\neq 4, and explain why cancelling the common factor does not assert…

APBC:9914005Evaluate Explain why multiplying by a conjugate is valid here, and explain why cancelling the common factor x-3 doe…

APBC:9914006Evaluate Show the algebraic simplification that is valid for x\neq 0.

APBC:9914007Evaluate using an explicit inequality and the Squeeze Theorem.

APBC:9914008Evaluate using the standard limit \lim_{u\to 0}(\sin u)/u=1.

APBC:9914009Evaluate using the identity 1-\cos\theta=2\sin^2(\theta/2) and the standard limit \lim_{u\to 0}(\sin u)/u=1.

APBC:9914010Evaluate, or determine that it does not exist, Check that the expression has real domain points on both sides of zero.

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APBC:9914001On the interval [-2,6], the graph of a function r consists of three line segments: the segment joining (-2,5) to (0…

APBC:9914002Selected values of a function g are given in the table below.

APBC:9914003Suppose Using only limit laws, evaluate

APBC:9914004Evaluate Show the factoring that is valid for x\neq 4, and explain why cancelling the common factor does not assert…

APBC:9914005Evaluate Explain why multiplying by a conjugate is valid here, and explain why cancelling the common factor x-3 doe…

APBC:9914006Evaluate Show the algebraic simplification that is valid for x\neq 0.

APBC:9914007Evaluate using an explicit inequality and the Squeeze Theorem.

APBC:9914008Evaluate using the standard limit \lim_{u\to 0}(\sin u)/u=1.

APBC:9914009Evaluate using the identity 1-\cos\theta=2\sin^2(\theta/2) and the standard limit \lim_{u\to 0}(\sin u)/u=1.

APBC:9914010Evaluate, or determine that it does not exist, Check that the expression has real domain points on both sides of zero.